OBSERVATIONAL 
GEOMETRY 

^-..-.•^••n,,...,.,.. • V;M...M • w •» ™y 


UC-NRLF 


WILLIAM -T- CAMPBELL 


PH1LLIPS-LOOMIS  MATHEMATICAL    SERIES 


OBSERVATIONAL  GEOMETRY 


BY 
WILLIAM   T.  CAMPBELL,  A.M. 

INSTRUCTOR  IN  MATHEMATICS  IN  THE  BOSTON  LATIN  SCHOOL 


WITH    AN    INTRODUCTION    BY 


ANDREW    W.  PHILLIPS,   PH.D. 

PROFESSOR     OF     MATHEMATICS     AND     DEAN     OF     THE 
GRADUATE   SCHOOL,  YALE   UNIVERSITY 


OVER  300  ILLUSTRATIONS  AND  DIAGRAMS 


NEW  YORK  •  :  •  CINCINNATI  .  :  .  CHICAGO 
AMERICAN    BOOK   COMPANY 


Copyright,  1899,  by  HARPER  &  BROTHERS. 

All  rights  reserved. 

vv.  p.  3 


CONTENTS 


^  PAGB 

INTRODUCTION vii 

PART  I 

ELEMENTARY   FORMS   AND   CONSTRUCTION   OF   MODELS. 
SIMPLE   EXPERIMENTS   IN    MENSURATION 

Chapter  I.  The  cube.  Squares ;  right  angles ;  construction  of  a  dia- 
gram ;  cutting  out  a  diagram ;  horizontal  surfaces ;  parallel 
faces;  vertical  planes  ;  a  test  of  geometric  equality  ;  the  three 
dimensions  in  geometry;  area  of  a  square,  volume  of  a  cube  .  I 

IT.  A  parallelepiped.  Construction ;  description  ;  quadrilat- 
erals ;  straight  lines  and  their  measurement ;  area  of  a  rect- 
angle;  volume  of  a  parallelepiped;  a  practical  experiment  in 
volumes 17 

III.  A  prism.      Construction;    description;  variety  in   prisms; 
triangles 31 

IV.  Angles.    Construction  and  measurement  of  angles  with  the 

aid  of  a  protractor 37 

V.  Construction  of  some  plane  figures.  Triangles;  sum  of 
the  angles  of  a  triangle  ;  a  right  triangle ;  parallel  lines  ;  paral- 
lelograms   45 

VI.   A  truncated  prism.     Construction ;  description     ....      55 

VII.    A   pyramid.      Construction  ;    description  ;  diedral  angles ; 

area  of  a  triangle  ;    volume  of  a  pyramid 58 

VIII.   A  triangular  pyramid.     Construction;  solid  angles    ...      65 

IX.  A  pentagonal  pyramid.     Construction 68 

X.   A  hexagonal  pyramid.     Construction 7° 


25972 


CONTENTS 

PAGE 

Chapter  XI.  Polygons  and  symmetry.  Varieties  of  polygons; 
symmetry  with  respect  to  a  line ;  symmetry  with  respect 
to  a  point ;  perimeters ;  diagonals ;  names  of  polygons ; 
distortion  of  polygons 72 

XII.   A  frustum  of  a  pyramid.     Construction;  description  .      Si 

XIII.  A  truncated  pyramid.     Construction ;  description       .      84 

XIV.  Curved  surfaces   and  lines.       The   circle;    railroad 
curves;  three  ways  of  drawing  a  circumference    ....       87 

XV.  A  cylinder.  Construction;  description;  length  of  a  cir- 
cumference ;  area  of  a  circle;  area  of  the  surface  of  a 
cylinder ;  volume  of  a  cylinder 97 

XVI.   A  cone.     Construction;  description;  area  of  the  surface  ; 

volume 102 

XVII.   Solids  of  revolution.     The  sphere;  description;  area 

of  the  surface  ;  map  drawing ;  volume 108 

XVIII.  Figures  for  practice.  General  questions  ;  a  truncated 
triangular  prism  ;  two  quadrangular  prisms  ;  the  regular 
octaedron ;  the  regular  icosaedron ;  the  regular  dodecae- 
dron  ;  a  pentagonal  prism ;  three  crystal  forms  .  .  .  .  116 


PART   II 

POINTS,   LINES,   ANGLES,   POLYGONS,    AND    CIRCLES. 

CONSTRUCTIONS,    MENSURATION,   SIMILAR 

FIGURES,   AND   SURVEYING 

Chapter  XIX.   Points  and  straight  lines.     Arrangements 131 

XX.  Points  of  intersection.     Made  in  various  ways  by  two 

groups  of  straight  lines 137 

XXI.   Angles.     Formed  by  two   lines  ;  by   three  lines   at   one 

point,  two  points,  and  three  points 145 

XXII.   Triangles.     Construction  of  various  kinds  .....     149 
Quadrilaterals.     Construction  of  various  kinds. 
Polygons.     Description ;  sum  of  the  angles ;  pentagons  ; 
hexagons. 

XXIII.  Circles.      Arrangements  of  two   circles ;   chords ;   arcs ; 
tangents;  secants 157 

XXIV.  Regular    polygons.     Construction ;   calculation   of  the 
length  of  the  circumference  of  a  circle 165 


CONTENTS  v 

PACK 

Chapter  XXV.  Constructions.  Straight  lines  ;  bisecting  straight  lines ; 
perpendiculars  ;  arcs  of  given  size ;  angles  of  given  size  ; 
bisecting  arcs  and  angles ;  circumscribed  and  inscribed 
circles;  miscellaneous  problems 17° 

XXVI.   Areas.     Rectangle  ;  parallelogram  ;   triangle  ;  trapezoid ; 

polygons ;  circle  ;  sector ;  segment ;  sphere 183 

XXVII.  Volumes.  Cube  ;  parallelepiped  ;  prism  ;  cylinder;  pyra- 
mid; cone;  sphere;  irregular  figures 199 

XXVIII.  Ratio  and  proportion.  Ratios  between  two  lines ;  pro- 
portions among  four  lines  ;  means  and  extremes ;  to  com- 
plete a  proportion;  to  divide  a  straight  line  into  equal 
parts 2I° 

XXIX.   Similar  figures.     Similar  polygons ;  triangles ;  construc- 
tion ;  areas  ;  similar  polyedrons ;  volumes    217 

XXX.   Surveying.    Instruments;  problems •    •    225 


INTRODUCTION 


IN  the  works  of  nature  and  of  man  Geometry  plays  a  most 
important  role.  The  rays  of  light  from  the  sun  suggest  the 
straight  line;  the  surface  of  still  water,  the  plane;  the  faces 
of  crystals,  a  variety  of  elementary  plane  figures  bounded  by 
straight  lines ;  while  the  crystals  themselves  suggest  the 
most  common  figures  bounded  by  planes.  Moreover,  the 
myriad  other  forms  in  the  animal,  the  vegetable,  and  the 
mineral  kingdoms  furnish  unending  variety  of  symmetrical 
and  complex  geometric  forms,  while  the  creations  of  the 
artist  and  the  architect,  and  the  problems  of  the  engineer 
and  the  astronomer,  all  have  their  basis  in  Geometry. 

The  practice  of  training  pupils  early  in  observing  the  sim- 
ple geometric  forms  and  relations  of  the  objects  which  come 
under  their  every-day  notice,  of  teaching  them  the  use  of  the 
simplest  tools  of  geometric  construction,  and  of  making 
them  familiar  with  a  variety  of  means  of  finding  lengths, 
areas,  and  volumes,  is  a  most  natural  and  potent  means  of 
training  their  powers  of  observation,  and  at  the  same  time 
of  cultivating  habits  of  concentrated  and  continuous  atten- 
tion. 

The  old  arithmetics  with  their  puzzling  problems  furnished 
a  powerful  means  for  the  cultivation  of  the  powers  of  anal- 
ysis, but  they  did  not  furnish  in  any  adequate  sense  the  care- 
ful training  of  the  child's  faculties  of  observation. 


viii  IN  '2  'MOD  UCT1ON 

It  is  true  that  many  of  their  problems  were  of  a  practical 
nature,  and  were  invaluable  as  a  means  of  familiarizing  the 
student  with  some  of  the  simple  rules  of  mensuration,  and  of 
creating  an  interest  in  the  methods  of  making  measurements 
for  obtaining  the  data  for  problems  to  which  these  rules 
might  be  applied — problems  in  rinding  the  contents  of  bins 
and  boxes,  and  of  calculating  the  amount  of  lumber  used  in 
their  construction ;  problems  in  rinding  the  areas  of  various 
shaped  fields ;  problems  in  finding  the  heights  of  trees  from 
their  shadows,  etc. 

Such  geometric  problems  often  awaken  an  intense  inter- 
est, and  a  desire  to  know  the  reasons  for  the  rules  employed 
in  their  solution,  and  so  create  an  appetite  for  the  study  of 
Formal  Geometry.  The  want, however,  of  careful  and  system- 
atic development  of  the  subject  as  a  means  of  cultivating 
the  faculties  of  observation  caused  a  revolt  against  the  arith- 
metic problems,  and  resulted  in  the  substitution  of  nature 
studies  to  a  considerable  extent  in  the  schools  for  the  drill  in 
such  problems.  But  nature  studies,  which  are  taught  mainly 
to  direct  attention  to  plant  and  animal  life,  and  to  the  mere 
observation  of  form,  fail  to  give  that  sharpness  to  the  men- 
tal faculties,  and  that  severe  training  in  vigorous  thinking, 
which  the  consideration  of  mathematical  problems  alone 
can  give. 

The  Observational  Geometry  combines  the  training  of  the 
nature  studies,  so  far  as  these  educate  the  eye  to  keen  and 
intelligent  perception,  with  the  training  which  the  more  val- 
uable problems  of  the  old  arithmetics  furnish,  and  so  gives 
a  mental  discipline  at  once  rigorous  and  entirely  free  from 
that  one-sidedness  which  either  of  these  systems  fosters 
when  taken  alone. 

It  gives  the  hand  dexterity  and  skill  in  making  drawings 
and  models  of  geometrical  figures.  It  trains  the  eye  to  esti- 
mate with  accuracy  forms  and  distances,  It  teaches  an  ap- 


IN  TROD  UCTION  ix 

preciation  of  beautiful  and  symmetric  forms.  It  seeks  out 
and  appropriates  methods  of  accomplishing  geometrical 
results  from  every  source  in  nature  and  every  employment 
in  life.  It  is  the  best  stimulant  for  the  inventive  faculties. 
It  makes  the  student  familiar  with  many  of  the  terms  and 
ideas  of  the  physical  sciences,  and  is  the  open  door  to  the 
successful  study  of  the  formal  and  the  higher  branches  of 
Geometry. 

ANDREW  W.  PHILLIPS. 


TO   THE   TEACHER: 

The  models  should  be  made  in  the  class-room,  under  the 
eye  of  the  teacher.  The  best  material  is  a  thin  card-board 
called  "  light  tag  stock,"  which  is  cheap,  and  can  be  procured 
in  quantities  cut  in  sheets  of  convenient  size.  The  pupil 
should  preserve  the  completed  models  in  a  box,  which  can 
serve  also  as  a  receptacle  for  drawing-materials.  The  ques- 
tion of  neatness  in  workmanship  should  be  settled  with  the 
first  model. 

The  number  of  models  to  be  made  will  vary,  of  course, 
with  different  classes  and  individuals,  as  also  will  the  work 
which  can  be  left  to  pupils  to  do  by  themselves ;  but  it  is 
intended  that  instruction  shall  be  largely  conversational. 
The  author  has  treated  the  cube  in  greater  detail  as  a  sug- 
gestion of  the  method  to  be  employed  with  other  figures. 

The  pupils  should  be  warned  that  dimensions  given  in  two 
systems  with  the  diagrams  are  alternatives  and  not  exact 
equivalents. 


FOR   REFERENCE 
MEASURES   OF  LENGTH,   WITH   EQUIVALENTS 


Metric  Table 


10  millimetres  (mm.)  = 

10  centimetres  = 

10  decimetres  = 

10  metres  = 

10  dekametres  =• 

10  hektometres  = 

10  kilometres  = 


centimetre  (cm.)  =  f  inches  nearly. 


decimetre 

metre 

dekametre 

hektometre 

kilometre 

myriametre 


=  3H 

=  391  " 
=  2  rods 
=  20    " 
=  |  miles 
=  6*    " 


The  metre  is  nearly  the  ten-millionth  part  of  the  distance  on  the  earth's 
surface  from  the  equator  to  either  pole,  first  calculated  in  France,  A.  D.  1799. 


1 2  lines  = 

1 2  inches  = 

3  feet  = 

5^  yards  =  i6£  feet  = 
40  rods  =  220  yards  = 

8  furlongs  =  5280  feet  = 


English  Table 

inch  (in.\ 
foot  (ft.) 
yard  (yd.) 
rod  (rd.) 


=  25  millimetres  nearly. 

=  3  decimetres        " 

=  0.9  metres  " 

=  5 

furlong  (fur.)  =  201      " 
mile  (m.)         =  1.6  kilometres      " 

The  yard  is  said  to  have  been  taken  from  the  length  of  the  arm  of  Henry  I. 
of  England,  A.  D.  noi. 

The  upper  edge  of  the  following  measure  is  one  decimetre  long,  and  is 
divided  into  centimetres  and  millimetres.  The  lower  edge  is  four  inches  long, 
and  is  divided  into  quarters  and  eighths  of  inches. 


1  1  IT 

1  1  1  1 

1  1  1  1 

III! 

III! 

Mil 

1 

1  1  1  !  1  1  1 

1  1  i  1  1  1  1  '1 

1  1  1  1 

i  j  1  1 

1  1  II 

lill 

!  1  1  1  |  i  I  1  1 

I  1  !  i  )  I  1  i  ! 

III  1 

nil 

1 

1 

1 

5 

6 

I 

1 
9 

2 

3 

4 

7 

8 

10 

1 

2 

3 

4 

•; 

I 

f 

1 

i 

1 

i 

i 

,1 

U.I          , 

i 

i 

.1-1     1  ... 

l 

i 

i 

PART    I 

ELEMENTARY  FORMS  AND  CONSTRUCTION  OF  MODELS 


SIMPLE  EXPERIMENTS  IN  MENSURATION 


OBSERVATIONAL  GEOMETRY 


CHAPTER   I 


THE    CUBE 

I.  WE  begin  to-day  the  study  of  geometry.  We  are 
going  to  model  and  study  some  of  the  principal  geometric 
figures.  The  picture  at  the  top  of  this  page  is  that  of  a  cube  ; 
you  have  seen  things  shaped  like  it,  —  blocks  of  stone,  glass 
paper-weights,  boxes,  and  occasionally  buildings,  or  at  least 
some  parts  of  buildings.  For  instance,  the  belfry  of  King's 
Chapel,  from  the  roof  of  the  portico  to  the  cornice,  is  a 
cube. 

The  sides  of  a  cube  are  all  alike ;   if  you  observe  the  one  in 
the  model  which  is  turned  toward  us,  you  will  see  that  its 


2  OBSERVATIONAL   GEOMETRY 

edges  are  all  straight,  all  of  the  same  length,  and  where  they 
come  together  at  the  corners  they  meet  "  straight  across  "  or 
perpendicularly,  so  that  the  corners  too  are  all  alike.  Now  in 
geometry  when  anything  has  four  straight  edges  and  four 


King's  Chapel,  Boston 

corners  like  this,  it  is   called  a  square.     You  will  remember 
that  we  are  speaking  of  only  one  side  of  the  cube. 

2.  How  to  draw  Right  Angles.  When  a  carpenter  wishes 
to  saw  a  piece  of  wood  straight  across,  or  to  make  a  square 
corner,  he  uses  that  flat  piece  of  steel,  which  you  may  have 


THE   CUBE 


seen,  called  the  "  carpenter's  square."  As  we  shall  have  to 
draw  ^quare  corners,  we  will  make  something  which  will  serve 
the  same  purpose  as  the  carpenter's  square. 


A  Carpenter's  Square 

Take  a  piece  of  rather  stiff  paper  about  the  size  of  a  sheet 
of  note-paper  opened  out,  fold  it  once,  then  turning  the  paper 
half  around  fold  it  again  straight  across  the  first  fold,  so  that 
the  edges  of  the  latter  will  come  just  evenly  against  each 
other.  If  you  have  done  this  correctly  you  will  find,  when 
you  open  the  paper,  that  there  are  two  straight  creased  lines 
crossing  each  other  straight  across  or  perpendicularly,  so  that 
the  corners  formed  by  these  crossing  lines  are  precisely  alike. 
When  straight  lines  come  together  as  these  do,  they  are  said 
to  meet  at  right  angles.  Now  fold  the  paper  up  again,  twice, 
as  before,  and  you  can  use  it  just  as  the  carpenter  uses  his 
square :  first,  however,  beginning  at  the  folded  corner,  along 


Right  Angles 

the  edge  where  the  paper  presents  a  single  fold,  make  an 
exact  copy  of  the  rule  given  on  the  page  containing  the 
table  of  measures  of  length, — metric  or  English,  whichever 
one  you  are  to  use  in  constructing  the  models. 


4  OBSERVATIONAL   GEOMETRY 

Since  we  now  know  something  about  a  square,  —  and  the 
sides  of  a  cube  are  all  squares,  as  you  will  remember,  —  we 
can  begin  to  make  a  model  of  the  cube. 

THE  CUBE;  CONSTRUCTION 


3.  How  to  draw  the  Diagram  for  a  Cube.  Take  a  piece  of 
cardboard  2  decimetres  5  millimetres  (or  81  inches)  long  and  i  decimetre 
6  centimetres  (or  6£  inches)  wide;  by  means  of  your  folded  measuring 
paper  see  that  the  bottom  and  left-hand  edges  are  straight  and  perpen- 
dicular to  each  other. 

Then,  beginning  at  the  bottom  of  the  paper,  at  A,  which  is  5  centi- 
metres 5  millimetres  (or  2^  inches)  from  the  left  edge,  draw  the  straight 
line  AB  (see  figure)  perpendicular  to  the  bottom  of  the  paper,  and  2 
decimetres  (or  8  inches)  long.  See  that  the  point  B  is  the  same  distance 
from  the  left  edge  as  the  point  A. 


THE   CUBE  5 

Then  beginning  again  at  the  bottom  of  the  paper,  at  C  which  is  5 
centimetres  (or  2  inches)  from  A,  draw  the  straight  line  CD,  also 
perpendicular  to  the  bottom  of  the  paper,  and  of  the  same  length 
as  AB.  See  that  D  is  5  centimetres  (or  2  inches)  from  B.  Divide 
the  lines  AB  and  CD  each  into  four  equal  parts  5  centimetres  (or 
2  inches)  long,  and  change  two  parts  —  the  third  counting  from 
the  bottom  —  into  dotted  lines.  Draw  BD  and  three  dotted  lines 
connecting  the  points  of  division  on  AB  and  CD.  These  four  lines 
should  be  perpendicular  to  AB  and  CD,  and  should  be  each  5  centi- 
metres (or  2  inches)  long. 

We  have  now  four  squares,  whose  sides  or  edges  are  all  of  the  same 
length,  and  whose  corners  are  all  "  square  "  or  right  angles. 

The  third  square  from  the  bottom  has  all  its  sides  dotted.  The  upper 
side  of  this  square  should  now  be  extended  to  E  and  F  by  straight  lines 
5  centimetres  (or  2  inches)  long  ;  the  lower  side  of  this  same  dotted  square 
should  be  extended  in  a  similar  way  to  G  and  H;  the  point  E  should  be 
connected  with  the  point  G,  and  the  point  /''with  the  point//.  There 
are  now  two  additional  squares,  both  of  which  should  be  tested  with  the 
folded  paper. 

4.  Dotted  Lines  and  Lapels.     In  the  figure  which  you  now  have, 
the  dotted  lines  are  for  folding.     Lapels  for  pasting  —  for  which  allow- 
ance should  be  made  when  you  cut  out  the  figure  —  will  come  on  the 
three  free  edges  of  the  top  square  and  on  the  outer  and  lower  edges  of 
the  two  squares  forming  the  arms  of  the  cross;  at  first  they  should  be 
made  5  millimetres  (or  \  inch)  wide,  but  after  practice  they  may  be 
made  narrower;  they  are  intended  to  come  inside  the  model. 

5.  What  is  a  Diagram?    You  have  now  drawn  what  is  called  a 
diagram  (dif-a-gram),  which  means  an  outline  or  sketch  .of  something. 
Your  diagram  represents  the  surface  of  a  cube.     A  diagram  may  or  may 
not  be  of  the  same  size  as  the  object  it  represents  ;  the  one  you  have 
drawn  is  of  the  same  size,  but  if  you  compare  it  with  the  one  on  page 
4,  you  will   see  how  the  two  differ,  although  they  represent  the  same 
thing.     The  diagrams  in  this  book  are  usually  smaller  than  the  objects 
themselves. 

6.  How  to  cut  out  a  Diagram.   Your  diagram  has  the  form  of  a  cross. 
It  may  now  be  cut  out  by  trimming  closely  the  border,  except  where  allow- 
ance must  be  made  for  the  lapels.     From  below  cut  close  up  to  the  lower 
corners  of  the  interior  dotted  square,  and  from  the  sides  cut  close  in  to  the 
two  upper  corners.     Trim  off  the  corners  of  the  lapels. 

With  the  aid  of  a  ruler  and  the  back  of  a  knife-blade  or  something  of  the 
kind,  you  should  crease  the  dotted  and  lapel  lines;  the  pencil-marks  will 
thus  come  inside  the  figure,  which  may  now  be  folded  and  pasted.  At 
first  you  are  likely  to  use  too  much  paste.  If  the  cardboard  is  rather 


6  OBSERVATIONAL   GEOMETRY 

thick,  you  will  4o  better  to  cut  the  folding  edges  half  through  with  a  knife 
and  let  the  pencil-marks  come  outside  the  figure;  then  probably  you 
would  have  to  use  glue  instead  of  paste.  You  can  produce  the  cleanest, 
sharpest  edges  by  laying  the  cardboard  on  a  glass  surface  and  by  using  a 
knife  instead  of  scissors,  guiding  the  knife  with  a  ruler.  L  is  the  last 
side  to  be  pasted. 


Testing  a  Plane  Surface 

7.   1.  How  many  sides  has  the  cube  ?     Of  what  shape  are  they  ? 

2.  How  many  edges  ? 

3.  How  many  corners  ? 

4.  Are  the  sides  flat  ?      To  test  whether  a  surface  is  flat,  hold  against  it  in 

several  positions  something  which  is  known  to  have  a  straight  edge  (like 
the  edge  of  a  ruler),  and  see  if  this  edge  touches  the  surface  all  along  its 
length :  if  it  does  so  in  all  positions,  the  surface  is  flat,  and  is  called  a 
plane.  The  word  "plane  "  is  derived  from  the  Latin  word  planus,  which 
means  "  flat."  Plane  surfaces  of  figures  are  also  called  faces,  which  is 
the  word  we  will  use  hereafter. 

5.  Have  any  objects  in  the  room  apparently  plane  surfaces?     Perhaps  you  can 

test  them  with  a  ruler. 

6.  How  many  edges  bound  each  face  of  the  cube  ? 

7.  Does  each  edge  form  a  part  of  the  boundary  of  more  than  one  face  ?     If  so, 

of  how  many  ? 


.    THE  CUBE  7 

8.  If  you  multiply  the  number  of  faces  by  the  number  of  edges  which  bound 

each  face,  by  what  must  you  divide  the  product  in  order  to   obtain  the 
number  of  different  edges? 

9.  How  many  corners  has  each  face  of  the  cube  ? 

10.  Does  each  corner  lie  in  more  than  one  face  ?     If  so,  in  how  many  ? 

11.  If  you  multiply  the  number  of  faces  by  the  number  of  corners  of  each  face, 

by  what  must  you  divide  the  product  in  order  to  obtain  the  number  of 
different  corners  ? 


Shuswap  Lake,  British  Columbia 


A  Plane  Surface 

8.  Horizontal  Surfaces.  Observe  now  the  surface  of  your 
desk  or  table,  and  see  if  there  is  any  part  on  which  objects 
would  not  slide  or  roll  of  themselves,  however  smooth  they 
and  the  desk  might  be.  If  so,  that  part  of  the  surface  of  the 
desk  is  horizontal.  The  horizon  is  the  line  where  the  sky  and 
the  earth's  surface  seem  to  come  together ;  and  a  horizontal 
plane  is  one  which  has  the  same  direction  as  a  plane  bounded 
by  the  horizon. 

The  surface  of  a  small  body  of  water  at  rest  is  horizontal, 
such  as  you  see  in  the  picture  of  the  lake.     A  test  whether 


8  OBSERVATIONAL   GEOMETRY 

a  certain  surface  is  horizontal  might  be  made  by  seeing  if 
all  parts  of  that  surface  could  touch  at  the  same  time  the 
surface  of  some  water  at  rest. 

12.  How  could  you  test  whether  the  top  of  your  desk  is  horizontal,  by  means 

of  a  glass  of  water  ? 

13.  How  would  you  describe  floors  and  ceilings  of  rooms  as  commonly  laid  ? 

14.  Do  you  know  of  any  floors  or  ceilings  in  your  school  building  which  are  not 

horizontal  ? 

15.  How  would  you  test  whether  a  string  drawn  tight  is  horizontal  ? 

9.  Parallel  Faces.  Now  place  the  cube  on  some  hori- 
zontal part  of  your  desk.  The  face  on  which  the  cube  rests 
is  called  the  base.  Is  the  base  of  the  cube  horizontal?  Is 
there  any  other  face  which  is  now  horizontal?  If  so,  those 
two  faces  are  parallel  (par'-al-lel)  to  each  other.  The  word 
parallel  is  derived  from  two  Greek  words  meaning  "  side  by 
side  one  another."  To  test  whether  two  faces  of  an  object 
are  parallel,  turn  the  object  so  that  one  of  the  two  faces  may 
become  horizontal :  then  if  the  other  face  is  also  horizontal, 
the  two  are  parallel  to  each  other.  Parallel  faces  cannot  meet 
one  another,  however  far  they  may  be  extended.  Moreover, 
parallel  faces  are  at  the  same  distance  apart  throughout  their 
extents.  In  the  case  of  the  cube  the  distance  between  the 
faces  can  be  measured  along  the  edges.  Measure  the  dis- 
tance between  the  two  faces  you  are  examining,  beginning  at 
each  of  the  four  corners  of  the  base.  If  you  find  a  variation  in 
the  four  results,  either  you  have  made  a  mistake  in  measur- 
ing, or  the  cube  was  not  accurately  constructed,  and  is  not 
really  a  cube. 

In  practice  carpenters  lay  floors  horizontally  by  the  aid  of 
certain  instruments,  of  which  a  common  example  is  the  spirit- 
level.  This  consists  of  a  straight  bar  of  wood,  in  the  upper 
side  of  which  is  a  glass  tube  slightly  curved  and  nearly  filled 
with  alcohol.  When  the  bottom  of  the  bar  is  horizontal,  a 
bubble  appears  exactly  in  the  middle  of  the  tube. 


THE   CUBE  9 

10.  Vertical  Planes.  Vertical  is  the  opposite  to  hori- 
zontal. It  is  the  direction  taken  by  a  plumb  line,  which  is  a 
cord  held  at  one  end,  hanging  motionless,  and  drawn  tight  by 
a  weight  (usually  a  piece  of  pointed  lead)  attached  to  the 
lower  end. 


Testing  a  Surface  with  a  Spirit  Level 

To  test  whether  a  plane  is  vertical,  a  plumb  line  is  hung 
near  it.  If  the  line  can  hang  freely  close  to  the  plane,  but 
without  touching  it,  the  plane  is  vertical. 

We  will  now  examine  the  four  side  faces  of  the  cube,  to 
compare  their  directions  with  that  of  the  base.  Placing  the 
cube  with  its  base  horizontal,  as  before,  observe  that  the 
direction  of  each  of  these  faces  to  the  base  is  the  same  as 
that  of  a  plumb  line  to  the  base.  So  each  of  these  faces  is 
vertical. 

The  side  faces  are  also  called  perpendicular  to  the  base. 
Two  planes  are  perpendicular  to  each  other  when  they  meet 


io  OBSERVATIONAL   GEOMETRY 

at  right  angles,  so  that  if  the  object  were  turned  about  and 
either  of  the  planes  were  placed  in  an  horizontal  position, 
the  other  would  become  vertical. 


A  Plumb  Line  and  a  Vertical  Rod 


16.  Among  the  four  vertical  faces  of  the  cube  are  there  any  which  are  perpen- 

dicular to  each  other  ?    Test  them  by  turning  the  figure  over  so  that  one 
of  the  two  may  be  horizontal. 

17.  What  is  the  direction  of  the  ceiling  of  your  room  ? 

18.  Of  the  walls  ? 

19.  Of  the  floor  ? 

20.  Is  the  ceiling  parallel  to  any  of  these  ? 

21.  Is  the  ceiling  perpendicular  to  any? 

22.  Are  any  of  the  walls  parallel  to  other  walls  ? 

23.  Are  any  of  the  walls  perpendicular  to  other  walls  ? 

24.  Is  the  door  vertical  or  horizontal  ? 

25.  Does  your  answer  to  the  previous  question  depend  upon  whether  the  door 

is^open,  shut,  or  ajar? 

26.  When  the  door  swings  on  its  hinges,  does  its  direction  change  with  refer 

ence  to  the  ceiling  ? 

27.  With  reference  to  its  own  wall  ? 

28.  With  reference  to  the  other  walls  ? 


THE   CUBE  II 

29.  Can  you  hold  a  book  open  so  that  one  cover  may  be  perpendicular  to  the 

other  cover  and  both  may  be  vertical  ? 

30.  Can  you,  so  that  one  cover  may  be  perpendicular  to  the  other  cover,  and 

neither  be  vertical  ? 

31.  Can  you  do  the  same  and  have  one  cover  horizontal  ?     If  so,  what  is  the 

direction  of  the  other  cover  ? 

32.  How  would  you  distinguish  between  vertical  and  perpendicular? 

33.  Between  horizontal  and  parallel  ? 


Tracing  the  Base  of  a  Cube 

ii.  A  Test  of  Geometric  Equality.  Next  we  will  examine 
the  shapes  and  compare  the  sizes  of  the  six  faces.  Place  the 
cube  on  a  blank  sheet  of  paper,  one  face  directly  before  you, 
and  with  a  pencil  trace  the  outline  of  the  base.  Then,  with- 
out lifting  the  cube,  turn  it  around  so  that  another  face  may 
be  before  you,  and  make  another  tracing  of  the  base  in  its 
new  position,  directly  over  the  first.  Turn  the  cube  twice 
more,  and  make  two  more  tracings. 


12  OBSERVATIONAL   GEOMETRY 

With  accurate  tracings  and  a  true  cube  the  four  tracings 
will  look  like  only  one.  Likewise,  if  you  turn  the  cube  over 
upon  any  other  face,  you  will  find  that  you  can  trace  its 
outline  exactly  over  the  first  outline,  and  in  four  different 
positions. 

34.  How,  then,  do  the  six  faces  compare  with  each  other  in  shape? 

35.  How  do  the  six  faces  compare  in  size  ? 

36.  How  many  edges  bound  each  face  ? 

37.  If  you  multiply  the  number  of  edges  of  each  face  by  the  number  of  faces, 

will  the  product  be  the  number  of  edges  of  the  cube  ?     Explain  your 
answer. 

38.  Do  the  two  edges  at  each  corner  of  each  face  extend  out  from  each  other  in 

the  same  way  ? 


On  the  Thames 

39.  Is  it  in  the  same  way  as  a  string  drawn  tight  horizontally  would  extend 

from  a  plumb  line  hanging  over  one  end  ? 

40.  Are  the  edges  all  of  the  same  length  ? 

41.  Are  the  faces  all  squares  ? 

42.  Describe  a  cube  by  its  faces,  giving  their  number,  shape,  and  relative  size. 

43.  How  many  faces  of  a  cube  are  parallel  to  any  one  face  ? 

44.  How  many  faces  are  perpendicular  to  any  one  face  ? 

45.  How  many  edges  are  parallel  to  any  one  edge  ? 

46.  How  many  edges  meet  any  one  edge  perpendicularly  ? 

47.  Can  you  hold  your  cube  so  that  eight  edges  may  be  horizontal  ? 

48.  So  that  only  four  edges  may  be  horizontal  ? 

49.  So  that  no  edge  may  be  horizontal  ? 

50.  So  that  four  edges  may  be  vertical  ? 

51.  So  that  no  edge  may  be  vertical  ? 

52.  The  accompanying  picture  represents  a  crew  ready  to  row  on  the  Thames. 

How  many  parallel  lines  do  you  see  ? 

53.  If  the  crew  keep  time  as  they  row,  will  these  lines  continue  to  be  parallel  ? 


THE   CUBE  13 

12.  The  Three  Dimensions  in  Geometry.  When  you  meas- 
ured the  distance  between  the  base  and  upper  surface  of  the 
cube,  you  were  measuring  one  dimension  of  the  cube,  which 
is  called  its  thickness,  or  heiglit,  or  depth. 

54.  In  what  instances  would  you  naturally  speak  of  the  thickness  of  objects? 

55.  Of  the  height  ? 

56.  Of  the  depth  ? 

Now  place  the  cube  as  before,  horizontally,  with  one  face 
directly  in  front  of  you.  You  will  see  that  two  faces  on  the 
sides  extend  away  from  you,  but  are  parallel  to  each  other. 
The  distance  between  these  two  faces  is  called  the  length  of 
the  cube.  Measure  the  distance  in  centimetres  or  in  inches. 

Lastly,  there  is  a  face  in  the  rear,  parallel  to  the  one  in 
front;  and  the  distance  between  these  two  faces  is  called  the 
breadth  or  width  of  the  cube.  Measure  the  distance  in  centi- 
metres or  in  inches. 

You  have  now  measured  the  three  dimensions  of  the  cube, 

—  length,  breadth,  and  thickness;   and  if  you  have  measured 

correctly,  and  if  the  figure  was  accurately  made  (that  is,  if 

it  really  is  a  cube),  you  have  found  that  all  three  dimensions 

of  the  cube  are  equal  to  each  other. 


13.   Areas.    We  will  now  examine  the  size  of  the  faces. 

Draw  on  paper  a  square  with  edges  5  cm.  long.     Divide 

each  edge  into  parts  I  cm.  long,  and  draw  lines  to  connect 

all  the  opposite  points  of  division;  some  of  the  lines  are 

shown  in  the  annexed  figure. 


OSSER  VA  TIONA  L   GE  OME  TR  Y 


57.  What  is  the  shape  of  the  parts  into  which  you  have  divided  your  square  ? 

58.  Into  how  many  parts  have  you  divided  it  ? 

Draw  a  square  With  edges  3  cm.  long ;  draw  dividing  lines 
as  before,  and  count  the  number  of  parts  into  which  the 
square  is  divided. 

Do  the  same  with  a  square  having  an  edge  of  4  cm. 

In  these  questions  you  have  been  finding  the  area  of  squares. 
The  area  of  a  square  means  the  number  of  smaller  squares 
into  which  it  can  be  divided. 


Square  Cm. 


Square  Inch 


If  each  edge  of  one  of  the  smaller  squares  is  I  cm.  long, 
it  is  called  a  square  centimetre,  and  the  larger  square  is  said 
to  contain  so  many  square  centimetres. 

If  each  edge  of  one  of  the  smaller  squares  is  I  inch  long,  it 
is  called  a  square  inch,  and  the  larger  square  is  said  to  contain 
so  many  square  inches. 

If  the  square  were  quite  large,  like  the  floor  of  a  room,  it 
would  be  divided  into  squares  having  an  edge  I  metre,  I  yard, 
or  I  foot  long,  the  smaller  squares  being  called  square  metres, 
square  yards,  or  square  feet. 

Can  you  give  a  rule  for  calculating  the  area  of  a  square 
without  actually  dividing  the  square  into  smaller  ones  when 
you  know  the  length  of  one  edge? 

Calculate  the  area  of  the  entire  surface  of  your  cube. 

In  finding  areas  you  have  not  considered  the  question  of  thickness; 
for  surfaces  have  only  two  dimensions,  — length  and  breadth.  They  have 
no  thickness,  being  merely  the  outsides  of  figures. 


THE  CUBE 


14.  Volumes.  Let  us  now  examine  .the  size  of  the  cube. 
If  your  cube  were  solid  and  yet  made  of  something  which 
could  be  cut  easily,  and  each  edge  were  divided  into  five 
equal  parts,  the  cube  could  be  cut  into  layers,  and  each  layer 
could  be  cut  into  a  number  of  small  cubes. 


59.  Can  you  see  how  many  layers  there  would  be  ? 

60.  Can  you  see  how  many  small  cubes  there  would  be  in  each  layer? 

61.  Can  you  calculate  how  many  small  cubes  there  would  be  in  the  whole 

figure  ? 

Each  of  these  smaller  cubes  is  called  a  cubic  centimetre, 
which  means  a  cube  whose  edges  are  one  centimetre  long. 
There  are  a  number  of  cubic  centimetres  indicated  in  the 
figure,  but  you  would  not  find  great  difficulty  in  making 
yourself  a  cubic  centimetre  out  of  paper,  using  the  diagram 
at  the  head  of  this  chapter. 

62.  How  many  cubes  of  the  size  you  have  made  would  it  take  to  form  a  cube 

with  edges  twice  as  long  ?     Perhaps  you  can  collect  the  cubes  from  your 
classmates,  and  try  the  experiment  by  placing  them  together. 

63.  How  many  cubes  would  it  take  to  form  a  cube  with  an  edge  three  times  as 

long  as  the  first  cube  ? 

64.  How  many  cubic  centimetres  are  there  in  a  cube  whose  edge  is  2.  cm.  long? 

65.  How  many,  if  the  edge  is  3  cm.  long  ? 


i6 


OBSERVATIONAL   GEOMETRY 


In  these  questions  you  have  been  finding  the  volumes  of 
cubes.  The  volume  of  a  cube  is  the  number  of  cubic 
centimetres,  metres,  inches,  feet,  etc.,  into  which  it  could  be 
divided. 

66.  Can  you  give  a  rule  for  calculating  the  volume  of  a  cube  when  you  know  its 
dimensions  ? 

The  area  of  a  square  is  the  length  of  an  edge  multiplied  by  itself. 

Area  square  =  s  x  s. 

The  volume  of  a  cube  is  the  length  of  an  edge  multiplied  by  itself  twice. 
Volume  cube  =  s  x  s  x  s. 


3| 


P»sta  Scissors    Dividers  Triangles        Compasses    Pencil      Parallel  Ruler 

Eraser  Protractor  Knife 

Graduated  Ruler 


CHAPTER    II 


A    PARALLELOPIPED 

I.  THIS  figure  is  called  a  parallelepiped  (par-al-lel-o-pi'- 
ped),  a  word  which  means  "  having  flat,  parallel  surfaces." 
The  figure  has  six  faces  like  the  cube,  which  in  fact  is  one 
kind  of  parallelepiped ;  but  the  name  is  usually  given  to 
figures  some  at  least  of  whose  faces  are  not  squares.  If  you 
observe  the  face  which  is  turned  towards  us,  you  will  see  that 
like  a  square  it  has  four  edges,  meeting  one  another  perpen- 
dicularly; but  unlike  those  of  the  square  the  edges  are  of 
two  different  lengths,  the  opposite  ones  being  equal.  Such  a 
face  is  called  a  rectangle,  which  means  "  hav- 
ing right  angles." 

Draw  a  square  with  an  edge  of  any  length,  say  5  cm. 
(or  2  in.)  and  cut  it  out  from  the  paper.  With  the  aid 
of  your  folded  measuring  paper,  rule  a  line  straight 
across,  perpendicular  to  the  edges  it  meets.  Then  cut 


i8 


OB  SEX  VA  1 1  ON  A  L   GE  OME  TR  Y 


Faneuil  Hall,  Boston 


Each  of 


the  square  into  two  parts  along  the  line  you  have  just  drawn, 
these  parts  will  be  a  rectangle. 

You  will  notice  that  the  opposite  edges  of  each  rectangle  are  parallel ; 
and  if  you  fold  the  rectangle  over  so  that  the  opposite  edges  may  come 
together,  you  will  see  that  they  are  equal. 

Also  you  can  cut  each  rectangle  into  smaller  rectangles,  being  careful 
to  make  the  dividing  lines  perpendicular  to  the  edges  they  meet ;  and  you 
can  turn  a  rectangle  into  a  square  by  cutting  off  the  right  amount. 


A    PARALLELOPIPED  19 

The  parallelepiped  is  by  far  the  commonest  of  all  forms  in 
architecture,  occurring  repeatedly  in  the  parts  of  buildings. 
The  picture  of  Faneuil  Hall,  for  example,  shows  five  distinct 
parallelepipeds,  —  three  in  the  body  of  the  building,  one  in 
the  chimney,  and  one  in  the  base  of  the  cupola.  All  the  faces, 
except  two  in  the  cupola,  are  rectangles;  so  we  have  here 
five  rectangular  parallelepipeds. 

We  will  now  make  a  model  of  the  parallelepiped. 


2.  The  diagram  will  need  paper  25  cm.  5  mm.  X  21  cm.  (or  10^  X  8|  in.). 
AB  and  CD  are  each  2  dcm.  5  cm.  (or  10  in.)  long  and  I  dcm.  (or  4  in.)  apart ; 
that  is,  AC  and  BD  are  each  I  dcm.  (or  4  in.)  long. 

AB  and  CD  are  each  divided  into  parts  beginning  at  A  and  C  as  follows: 
5  cm.  (or  2  in.),  7  cm.  5  mm.  (or  3  in.),  5  cm.  (or  2  in.),  and  7  cm.  5  mm.  (or  3  in.). 

jEFand  GH are  each  2  dcm.  (or  8  in.)  long,  extending  at  each  end  5  cm.  (or 
2  in.)  beyond  AB  and  CD,  which  they  cross  at  the  first  and  second  points  of 
division  from  A  and  C.  EG  and  FH  are  each  7  cm.  5  mm.  (or  3  in.)  long. 

Wherever  the  lines  cross  they  are  perpendicular  to  each  other. 
3    1.  How  many  faces  has  this  figure? 

2.  How  many  edges  ? 

3.  How  many  corners  ? 

4.  If  you  place  the  figure  with  any  face  horizontal,  will  any  other  faces  also  be 

horizontal  ?     If  so,  how  many  ? 


20 


OBSERVATIONAL    GEOMETRY 


5.  What  other  name  could  be  given  to  those  faces,  comparing  their  directions 

with  each  other  ? 

6.  If  the  base  were  horizontal,  would  any  faces  be  vertical  ?    If  so,  how  many  ? 

What  other  name  could  be  given  to  those  faces,  comparing  their  direc- 
tions with  that  of  "the  base? 

7.  Is  it  true  that  each  face  of  this  figure  is  bounded  by  two  pairs  of  parallel 

edges  ? 

8.  Is  it  true  that  the  edges  of  each  face  which  meet  are  perpendicular  to  each 

other  ? 

9.  How  would  you  answer  the  last  two  questions  in  the  case  of  the  faces  of 

the  cube  ? 

10.  Are  the  faces  of  the  new  figure  squares  ?     If  not,  what  difference  do  you  see 
between  them  and  squares  ? 


Square 


Rhombus 


Rectangle 


Rhomboid 


4.  Quadrilaterals.  Parallelograms.  Any  face  which  is 
bounded  by  four  edges  is  called  ^quadrilateral  (<^\\&&-n-\2,\! -zv- 
al),  which  means  "  four-sided."  The  square  and  the  rectan- 
gle, however,  belong  to  a  particular  class  of  quadrilaterals, 
called  parallelograms  (par-al-lel'-o-grams),  which  means 
"parallel  marks  or  lines."  You  will  remember  that  the 
angles  of  the  square  and  rectangle  are  right  angles :  now,  if 
in  these  figures  you  change  the  direction  of  two  opposite 
sides  as  regards  the  other  two,  there  will  no  longer  be  any 
right  angles,  but  there  will  be  two  acute  and  two  obtuse 
angles  in  each  figure.  This  is  what  is  called  "  distorting"  a 
figure.  If  you  distort  the  square  and  the  rectangle  you 


A  PARALLELOPIPED 


21 


form  two  othev  parallelograms  :  from  the  square  you  form 
the  rhombus  (rhom'-bus),  and  from  the  rectangle  you  form 
what  was  formerly  called  the  r/tomfeid  (thorn* -boid),  though 
this  term  is  no  longer  in  general  use,  and  the  figure  is  com- 
monly known  simply  as  a  parallelogram.  When  no  variety 
is  specified,  "  parallelogram "  is  understood  to  mean  the 
rhomboid. 

A  rhombus  is  a  parallelogram  whose  edges  are  all  equal, 
but  whose  angles  are  not  right  angles. 

The  word  means  "  something  which  can  be  whirled  around,"  the  shape  having 
some  likeness  to  an  old-fashioned  whirling  spindle. 

The  rhomboid  and  rhombus  can  be  formed  from  the  rectangle  by  cutting. 


Draw  a  rectangle  A  BCD  with  edges  7  cm.  and  4  cm.  (or  3^  ki.  and 
2  in.)  and  cut  it  out  from  the  paper. 

Beginning  at  two  opposite  corners,  A  and  C,  measure  on  opposite 
edges  the  equal  lengths  A X  and  CY  3  cm.  (or  ii  in.);  and  draw  the 
lines  DX  and  BY.  Then  cut  through  the  lines  DX  and  BY.  The  part 
left,  DYJ3X,  is  a  rhomboid.  You 'will  see  that  the  opposite  edges  are 
parallel ;  and  by  measuring  the  lengths  you  will  find  that  the  opposite 
edges  are  also  equal.  If  you  do  the  work  accurately,  the  lengths  will 
prove  to  be  4  cm.  and  5  cm.  (or  2  in.  and  2\  in.). 

Then,  beginning  at  the  ends  of  one  of  the  shorter  edges,  measure  on 
the  longer  edges  XE  and  Z^each  i  cm.  (or  1  in.)  long;  draw  the  line 
EF\  and  cutting  along  this  line  divide  the  rhomboid  into  two  parts. 
The  smaller  of  the  two  parts  will  be  another  rhomboid.  The  greater 
part,  EFYD,  will  be  a  rhombus. 

All  four  forms  of  the  parallelogram  agree  in  having  the 
opposite  edges  parallel;  and  all  have  the  opposite  edges 
equal. 


22  OBSERVATIONAL   GEOMETRY 

In  what  particular  respect  does  a  rectangle  resemble  a  square  ? 

In  what  particular  respect  does  a  rhombus  resemble  a  square? 

In  what  does  a  rectangle  differ  from  a  square? 

In  what  does  a  rhombus  differ  from  a  square  ? 

Take  a  piece  of  string,  tie  three  knots  in  it,  and  lay  it  down  on  your  desk  in 
the  form  of  a  square,  the  knots  coming  at  the  corners.  Then  change  the  square 
into  a  rhombus  having  the  same  knots  at  the  corners. 

Lay  the  same  string  down  in  the  form  of  a  rectangle  with  knots  at  the 
corners.  Will  the  knots  be  the  same  as  you  used  for  the  square? 

Can  you  change  the  rectangle  into  a  parallelogram  without  making  new 
knots  ? 


Trapezoid 

There  are  two  more  forms  of  plane,  surfaces  bounded  by 
four  edges, —  the  trapezoid  (trap'-e-zoid)  and  trapezium  (trap- 
e'-zi-um). 


Trapezium 

A  trapezoid  has  two  edges  which  are  parallel  and  two  which 
are  not  parallel. 

The  word  means  "  like  a  table." 

A  trapezium  has  none  of  its  edges  parallel. 

The  word  means  "  a  little  table." 

Do  you  see  how  to  change  a  parallelogram  into  a  trapezoid  by  cutting  it  once  ? 

How  many  times  must  you  cut  to  change  a  parallelogram  into  a  trapezium  ? 

In  the  following  collection  of  quadrilaterals,  judging  by  the 
eye,  give  the  name  of  each :  — 


A    PARALLELOPIPED 


32.  When  you  measured  the  cube,  what  did  you  find  about  the  three  dimensions  ? 

33.  Are  the  three  dimensions  of  your  rectangular  parallelepiped  equal  to  each 

other  ? 

34.  What  is  the  length,  —  that  is,  the  longest  dimension? 

35.  What  is  the  breadth  ? 

36.  What  is  the  thickness? 

37.  Can  you  place  the  figure  so  that  its  thickness  or  height  may  be  its  greatest 

dimension  and  its  length  may  be  its  smallest  dimension  ? 

38.  What  are  the  dimensions  of  the  two  greatest  faces  of  this  figure? 

39.  Of  the  next  greatest  ? 

40.  Of  the  smallest? 

41.  How  are  those  faces  which  are  equal  situated  with  respect  to  each  other? 

5.  Lines.  We  will  now  examine  the  edges  more  partic- 
ularly. Edges  are  lines  ;  furthermore,  they  are  the  only  true 
"lines"  in  the  geometric  sense  of  the  word.  A  "line"  in 
geometry  has  only  one  dimension,  —  length  ;  it  has  no  breadth 
or  thickness.  You  can  represent  a  line,  however,  by  making 
a  mark  on  a  surface  with  a  pen,  a  pencil,  or  even  a  crayon. 
The  boundaries  of  surfaces  are  lines:  wherever  two  surfaces 
meet  each  other,  there  is  a  line  in  common. 

A  straight  line  is  formed  where  two  plane  surfaces  meet. 
Thus  the  edges  of  cubes  and  parallelepipeds  are  all  straight 


24  OBSERVATIONAL   GEOMETRY 

lines.  The  word  straight  originally  meant  "  stretched,"  a 
cord  drawn  tight  representing  a  geometric  straight  line. 
Notice  that  a  straight  line  keeps  the  same  direction  through- 


A  Straight  Line 

out  its  length,  and  that  straight  lines  are  all  of  one  kind, 
though  we  sometimes  speak  of  a  "  broken  "  or  a  "  zigzag  " 
line,  which  is  really  a  collection  of  straight  lines. 


A  Broken  Line 


6.  The  length  of  a  straight  line  is  measured  by  applying 
to  it  any  unit  which  may  be  agreed  upon  as  convenient,  such 
as  a  centimetre,  metre,  kilometre,  inch,  foot,  mile.  For  short 
lines  an  inch  or  a  centimetre  is  a  convenient  unit;  for  long 
lines,  a  mile  or  a  kilometre. 

Two  sets  of  measurements  aje  in  common  use,  the  metric 
and  the  English.     Both  have  already  been  given. 


A   PARALLELOPIPED  25 

The  English  system,  being  in  common  use  in  the  United 
States,  is  likely  to  seem  easier;  but  with  practice  in  actual 
measurements,  the  metric  system  will  be  found  the  simpler 
to  use.  You  should,  however,  accustom  yourself  to  make 
measurements  in  each  system,  first  judging  by  the  eye,  and 
then  measuring  accurately  with  the  yard  or  metre  stick. 

42.  Judge  by  the  eye  the  lengths  of  the  following  lines,  and  then  test  them  by 
accurate  measurements:  — 


A  straight  line  is  the  shortest  which  can  be  drawn  between 
two  of  its  parts,  —  for  instance,  its  ends.  This  affords  one 
test  whether  a  given  line  is  straight  or  not;  for  if  the  shortest 
distance  between  the  ends  is  equal  to  the  length  of  the  line, 
then  the  line  is  straight.  Let  some  boy  hold  a  string  against 
the  blackboard  by  the  ends  so  that  it  may  sag  a  little.  Let 
another  boy  measure  the  distance  between  the  ends  with  a 
ruler  (the  edge  of  which  is  supposed  to  be  straight),  and 
compare  the  result  with  the  length  of  the  string. 

From  the  beginning  you  have  been  measuring  the  edges 
of  figures  as  if  you  knew  them  to  be  straight  lines.  This  was 


26  OBSERVATIONAL   GEOMETRY 

right;    for  plane  surfaces    always  form  straight  lines  where 
they  meet  or  cut  each  other. 


A B        A- 

I 2 


Lines  are  commonly  designated  by  two  letters  or  two  numbers,  one 
being  placed  at  each  end.  But  a  line  is  sometimes  designated  by  a 
single  letter  or  number  placed  anywhere  upon  it. 

7.  Suppose  you  wish  to  draw  a  straight  line  of  a  required 
length. 

If  the  length  of  the  line  you  are  to  draw  is  given  in  deci- 
metres or  in  inches,  you  can  draw  the  line  with  the  aid  of 
a  ruler  having  a  graduated  edge,  such  as  is  represented  on 
the  page  containing  the  table  of  measures  of  length.  It  is 
a  problem  you  have  done  since  you  began  this  book. 

If  the  length  of  the  required  line  is  not  given  in  numbers, 
but  is  shown  by  another  line  whose  length  in  numbers  is  not 
known,  you  can  perform  the  problem  in  either  of  two  ways. 

Suppose  you  are  required  to  draw  a  line  equal  in  length 
to  AB. 

A B 

First,  you  may  measure  the  length  of  AB  with  a  graduated 
ruler,  and  then  draw  another  of  the  same  length.  If  you  find 
AB  to  be  3  cm.  long,  you  have  only  to  draw  another  line 
3  cm.  long,  and  the  problem  will  be  performed.  This  way  is 
called  "  doing  the  problem  by  arithmetic."  The  difficulty 
is  that  the  length  of  AB  may  not  correspond  exactly  to  any 
distance  shown  by  your  graduated  scale.  The  next  method 
is  preferable. 

Secondly,  you  need  not  find  the  length  of  AB  in  numbers, 
but  instead  you  may  mark  on  a  strip  of  paper  which  has  a 
smooth  edge  two  dots  showing  the  length  of  AB  ;  and  then 


A    PARALLELOPIPED 


27 


having  drawn  a  line  of  any  length,  you  can  mark  ofif  on  it  the 
distance  shown  by  the  two  dots. 


Measuring  a  line  with  a  pair  of  dividers 

There  is  also  an  instrument  which  is  used  for  this  purpose, 
called  "  dividers,"  or  "  a  pair  of  dividers."  This  has  two 
prongs  opening  on  a  hinge,  so  that  the  distance  between  the 
pointed  ends  shows  the  length  of  a  line.  This  method  is 
called  "  doing  the  problem  by  geometry." 
43.  Draw  lines  equal  to  the  following  with  the  aid  of  a  graduated  ruler:  — 


28 


OBSERVATIONAL   GEOMETRY 


44.  Draw  lines  equal  to  the  following  "  by  Geometry  " :  — 


8.   Area  of  a  Rectangle.     Draw  on  paper  a  rectangle  10  cm. 
long  and  5  cm.  wide,  representing  one  of  the  faces  of  your 


parallelepiped.  Divide  the  edges  into  parts  i  cm.  long,  and 
draw  lines  connecting  the  opposite  points  of  division,  as 
indicated  in  the  diagram. 

45.  What  is  the  shape  of  the  parts  into  which  you  have  divided  the  rectangle  ? 

46.  Count  the  number  of  these  parts. 

47.  Can  you  say  that  there  are  ten  rows  with  five  in  each  row  ? 

48.  Is  it  also  true  that  there  are  five  rows  with  ten  in  each  row  ? 

49.  What  would  you  say  is  the  area  of  this  rectangle  ? 

Next  draw  on  paper  a  rectangle   IO  cm.  long  and    7  cm. 
5  mm.  wide,  representing  another  face  of  your  parallelepiped. 


A   PARALLELQPIPED 


29 


Divide  the  two  longer  edges,  AB  and  CD,  into  parts  I  cm. 
long.  Then,  beginning  at  A  and  B,  mark  off  on  AC  and  BD 
parts  I  cm.  long  as  far  as  they  will  go.  Draw  lines  as  before, 
connecting  the  opposite  points  of  division. 

50.  Count  the  number  of  whole  squares  thus  formed. 

51.  Count  the  number  of  the  parts  which  remain. 

52.  How  many  of  those  parts  would  it  take  to  make  one  of  the  squares  ? 

53.  How  many  squares  would  those  parts  make  if  cut  out  and  matched  together  ? 

54.  Can  you  say  that  there  are  ten  rows  with  seven  and  one-half  squares  in 

each  row? 

55.  What  would  you  say  is  the  area  of  this  rectangle? 

Lastly,  draw  a  rectangle  seven  and  one-half  centimetres 
long  and  five  centimetres  wide;  divide  it  into  squares  and 
parts  of  squares  as  before. 

56.  Count  the  number  of  squares. 

57.  Count  the  number  of  other  parts. 

58.  What  would  you  say  is  the  area  of  this  rectangle  ? 

59.  Can  you  give  a  rule  for  calculating  the  area  of  a  rectangle  when  you  know 

its  length  and  width  ? 

60.  Calculate  the  area  of  the  entire  surface  of  your  parallelepiped. 

9.   Volume  of  a  Parallelepiped.     The  volume  of  a  parallele- 
piped is  found  by  the  method  you  used  with  the  cube,  the 


.  .,. .-  + ..  +  .-  f.  .  .  r  . .  y.  __  4  --/--. 


figure  being  divided  into  small  cubes.  The  height  shows 
the  number  of  layers  of  cubes,  and  the  area  of  the  base  shows 
the  number  of  cubes  in  each  layer. 

61.  The  base  of  your  parallelepiped  is  10  cm.  long  and  ;£  cm.  wide.      How 
many  square  centimetres  are  there  in  its  area? 


30  OBSERVATIONAL   GEOMETRY 

62.  How  many  cubic  centimetres,  therefore,  are  there  in  one  layer  ? 

63.  The  height  is  5  cm.     How  many  layers  of  cubes,  therefore,  are  there  ? 

64.  What  is  the  total  number  of  cubic  centimetres  in  the  volume  of  the  figure  ? 

65.  Can  you  give  a  rule  for  calculating  the  volume  of  a  parallelopiped  when 

you  know  the  three  dimensions  ? 

10.  A  Practical  Experiment.  You  will  find  it  interesting  to  try 
practical  experiments  in  comparing  the  volumes  of  figures  which  you 
make.  Now,  as  the  edges  of  your  cube  are  5  cm.  long,  its  volume  is  125 
cubic  centimetres.  As  the  dimensions  of  your  parallelopiped  are  10,  7^, 
and  5  cm.,  its  volume  is  375  cubic  centimetres,  which  is  exactly  three 
times  the  volume  of  the  cube.  The  parallelopiped,  therefore,  ought  to 
hold  three  times  as  much  as  the  cube ;  and  you  can  test  this  by  filling 
the  cube  with  sand,  sawdust,  or  water,  etc.,  and  pouring  the  contents  into 
the  parallelopiped  until  the  latter  is  full.  For  this  it  will  be  better  to  have 
special  figures  with  one  side  open.  If  you  give  the  figures  a  coat  of  thick 
varnish,  inside  and  out,  they  will  hold  water. 

When  you  are  done,  preserve  the  two  figures  carefully ;  for  we  shall 
need  them  in  future  experimenting. 


Diagram  for  a  Measuring  Cube 

'  The  area  of  a  rectangle  is  the  product  of  its  two  dimensions. 

Area  rectangle  —  a  x  b. 

The  volume  of  a  parallelopiped  is  the  product  of  its  three  dimensions. 
Volume  parallelopiped  =  a  x  b  x  c. 


CHAPTER   III 


A    PRISM 

r.  THIS  figure  is  called  a  prism,  which  means  "some- 
thing sawed  off;  "  that  is,  a  prism  is  apart  of  another  figure; 
when  you  have  made  the  prism  you  will  see  what  is  the  figure 
of  which  it  is  a  part.  The  face  turned  towards  us  is  a  square; 
the  one  which  extends  to  the  rear  on  the  right  is  another 
square;  the  one  towards  the  left  is  a  rectangle. 

Each  of  the  faces  at  the  top  and  bottom 
is  a  triangle  (tri'-an-gle),  which  means 
" three-cornered." 

Draw  a  square  with  an  edge  of  5  cm.  (or  2  in.), 
and  cut  it  out  from  the  paper ;  draw  a  line  from 
corner  to  corner,  and  then  cut  the  square  into 


32  OBSERVATIONAL   GEOMETRY 

two  parts  along  this  line :  each  part  will  be  a  triangle  representing  the 
upper  and  lower  faces  of  the  prism. 

You  can  see  examples  of  triangular  prisms  in  the  two  dormer  windows 
on  the  roof  of  "  Shakespeare's  House."  If  you  imagine  a  plane  to  extend 
horizontally,  dividing  the  windows  into  two  parts,  the  lower  part  in  each 


Shakespeare's  House 

case  will  be  a  triangular  prism  resembling  the  model  given  above.  The 
bases  are  the  vertical  triangles  close  to  the  roof ;  they  are  still  called 
"  bases,"  although  the  prisms  here  do  not  rest  upon  them.  The  roof  of 
the  building  forms  the  rectangular  faces,  and  the  front  of  the  building 
and  the  imaginary  cutting  plane  the  square  faces. 

We  will  now  make  a  model  of  the  triangular  prism. 

2.  The  diagram  will  need  paper  18  X  15  cm.  (or  7^  X  6  in.).  AB  is  to  be 
I  dcm.  5  cm.  (or  6  in.)  long,  and  is  divided  into  three  equal  parts  at  D  and  G, 
5  cm.  (or  2  in.)  long. 

CE  and  /'T/are  each  i  dcm.  (or  4  in.)  long;  they  cross  AB  perpendicularly 
at  D  and  G,  where  they  are  divided  into  equal  parts. 


A    PRISM 


33 


When  so  much  has  been  finished,  AE  and  BH  should  be  drawn,  and  then 
CE  and  FH  should  be  prolonged  so  that  El  and  HJ  may  be  equal  to  AE 
and  BH. 

Lastly,  CV^and  IJ  should  be  drawn, 


10. 
11. 
12. 
13. 
14. 
15. 


1.  How  many  faces  has  this  figure  ? 
How  many  edges  ? 
How  many  corners? 

Are  there  any  parallel  faces  ?     If  so,  how  many  pairs  ? 
Are  there  any  parallel  edges  ?     If  so,  how  many  groups  ? 
What  is  the  greatest  number  of  parallel  edges  in  any  one  group  ? 
Are  there  any  edges  perpendicular  to  other   edges?     If  so,  what   is  the 

greatest  number  of  edges  which  meet  any  one  edge  perpendicularly  ? 
How  many  faces  are  there  each  bounded  by  four  edges  ?    Are  those  faces 

all  equal  to  each  other  ?     What  test  would  you  apply  ? 
How  many  edges  bound  each  of  the  other  faces  ?     Are  those  faces  equal  ? 

Test  them. 

Can  you  hold  the  figure  so  that  six  edges  may  be  horizontal  ? 
So  that  five  edges  may  be  horizontal  ? 
So  that  three  edges  may  be  horizontal  ? 
So  that  two  edges  may  be  horizontal  ? 
So  that  two  edges  may  be  vertical? 
So  that  three  edges  may  be  vertical  ? 


34  OBSERVATIONAL   GEOMETRY 

4.  Variety  in  Prisms.    This  figure  is  called  a  prism,  which, 
as  was  said  before,  means  "  something  sawed  off;  "    that  is, 
a  prism  is  a  part  of  another  figure. 

16.  Can  you  see  how  your  prism  is  a  part  of  a  cube  ?    Can  you  place  two  of  the 

prisms  together  so  as  to  make  a  cube  ? 

There  is  no  limit  to  the  varieties  of  prisms ;  but  all  prisms 
agree  in  having  two  faces  parallel  and  equal  to  each  other 
(these  faces  being  bounded  by  any  number  of  edges),  and 
all  other  faces  parallelograms. 

17.  Of  what  kind  or  kinds  are  the  parallelograms  in  your  prism  ? 

18.  The  parallelograms  may  or  may  not  be  parallel  to  each  other.     How  are 

they  in  your  prism? 

19.  The  parallelograms  may  or  may  not  be  equal  to  each  other.     How  are  they 

in  your  prism  ? 

The  parallelograms  are  called  the  lateral  (lat'-er-al)  or  side 
faces  of  the  prism  ;  "  lateral  "  means  "  on  the  side." 

The  two  faces  which  must  be  parallel  and  equal  are  called 
the  bases  of  the  prism  ;  and  prisms  take  various  names  ac- 
cording to  the  shape  of  their  bases,  —  rectangular,  square, 
triangular,  etc. 

A  right  prism  is  a  prism  whose  side  faces  are  all  squares 
or  rectangles;  here,  "right"  means  "straight." 

20.  Of  which  kind  is  your  prism  ? 

21.  Are  parallelopipeds  a  variety  of  prisms? 

22.  If  so,  how  many  pairs  of  the  faces  may  be  called  bases  ? 

23.  How  does  this  differ  from  the  case  of  other  prisms? 

24.  As  prisms  are  named  according  to  the  shape  of  their  bases,  what  kind  of  a 

prism  is  a  cube  ? 

25.  Is  the  cube  a  right  prism  ? 

26.  What  kind  of  a  prism  is  a  rectangular  parallelepiped  ? 

27.  Is  it  a  right  prism  ? 

5.  Triangles.     Let  us  now  examine  the  bases  of  the  prism 
you  have  just  made.     How  many  edges  bound  each? 

A  face  which  is  bounded  by  three  edges  is  called  a  triangle. 

There  are  several  kinds  of  triangles ;   but  all  can  be  formed 

by  cutting  quadrilaterals  into  two  parts  from  corner  to  corner. 


A   PRISM 


35 


An    eqttilateral   (e-qui-lat'-er-al)    triangle    is    bounded   by 
three   equal  edges. 

"  Equilateral  "  means  "  having  equal  sides." 


Equilateral 

An  isosceles  (i-sos'-ce-les)   triangle  has  two   of  its   edges 
equal. 

"  Isosceles  "  means  "  having  equal  legs,"  the  side  which  is  not  equal  to  the 
other  two  being  called  the  "  base." 


Isosceles 

A  scalene  (sca-Iene')  triangle  is  bounded  by  three  unequal 
edges. 

"  Scalene  "  means  "  having  crooked  legs." 


Scalene 


OBSERVATIONAL    GEOMETRY 


An  oblique  (o-blique')  triangle  has  none  of  its  edges  per- 
pendicular to  one  another.  It  may  be  equilateral,  isosceles, 
or  scalene.  The  preceding  are  examples  also  of  oblique 
triangles. 

A  right  triangle  has  two  of  its  edges  perpendicular  to  each 
other. 


Rteht  Scalene 


Right  Isosceles 


A  right  triangle  may  be  either  scalene  or  isosceles.  The 
edge  which  lies  opposite  the  right  angle  is  called  the  hypote- 
nuse (hy-pot'-e-nuse). 

In  the  following  collection  of  triangles,  give  the  name  of 
each,  first  judging  the  forms  by  the  eye,  and  then  testing 
them  by  measuring  the  sides. 


CHAPTER    IV 

ANGLES 

I.  NOTICE  the  hands  of  the  clock 
in  the  illustration.  The  hour  hand 
is  horizontal,  and  the  minute  hand 
is  vertical ;  the  two  are  therefore  at 
right  angles  with  each  other. 

As  the  hands  of  a  clock  move,  they 
are  at  right  angles  only  twice  during 
each  hour;  but  at  every  instant  they 
are  making  an  angle  of  some  kind 
with  each  other. 


An  Angle 

An  angle  is  a  figure  formed  by  two 
straight  lines  diverging  from  a  point. 

The  word  angle  is  derived  from  the  Latin 
word  angulus,  meaning  "a  corner." 


Belfry  of  Christ  Church,  Boston 


38  OBSERVATIONAL   GEOMETRY 

The  two  lines  are  called  the  sides  or  legs  of  the  angle. 
The  place  where  the  lines  meet  is  called  the  vertex  (ver'-tex) 
of  the  angle. 

Vertex  is  a  Latin  word  meaning  "  a  top  or  turning  point." 


Vertex 

The  vertex  is  a  point.  A  point  has  position  only,  —  no 
length,  breadth,  or  thickness. 

The  size  of  an  angle  depends  only  upon  the  amount  of 
inclination  of  one  side  to  the  other;  it  is  not  changed  by 
lengthening  or  shortening  the  sides.  The  hands  of  a  watch 
make  countless  different  sizes  of  angles  with  each  other  in 
the  course  of  an  hour,  but  do  not  change  their  own  length 
meanwhile ;  at  three  o'clock  and  nine  o'clock  the  hands  of  a 
great  clock  and  those  of  a  small  watch  are  alike  perpendicular 
to  each  other,  —  that  is,  are  at  right  angles. 


An  Acute  Angla 

An  acute  angle  is  less  than  a  right  angle. 

Acute  means  "  sharp." 


An  Obtuse  Angle 

An  obtuse  angle  is  greater  than  a  right  angle. 

Obtuse  means  "  blunt." 


ANGLES  39 

An  angle  may  be  designated  by  a  letter  or  number  placed  near  the 
vertex,  or  by  three  letters  or  numbers  placed,  one  at  the  vertex,  and  one 
on  each  side  of  the  angle. 


If  three  letters  are  used,  the  one  which  indicates  the  vertex  is  placed 
between  the  other  two  in  referring  to  the  angle,  as  BAC.  If  no  other 
angle  has  the  same  vertex,  an  angle  is  clearly  designated  by  one  letter; 
but  if  several  angles  have  the  same  vertex,  three  letters  are  generally 
used  so  as  to  avoid  confusion,  or  one  letter  can  be  placed  between  the 
sides  of  each  angle. 

,B 


2.   Table  of  the  divisions  of  the  right  angle. 
The  right  angle  is  divided  into  degrees  (°),  minutes  ('), 
and  seconds  ("),  according  to  the  following  table:  — 

60  seconds  (")  =  I  minute  (') 
60  minutes  (')  =  I  degree  (°) 
.     90  degrees  (°)  =  I  right  angle. 

1.  How  do  you  read  the  angle  18°  27'  43"  ? 

2.  85°  1 4' 30"? 

3.  60°  20' 48"? 

4.  Write  with  the  signs ;  ten  degrees,  forty  minutes,  twenty  seconds. 

5.  Thirty-eight  degrees,  seventeen  minutes,  six  seconds. 

6.  How  many  degrees  are  there  in  two-thirds  of  a  right  angle  ? 

7.  In  three-fourths  of  a  right  angle  ? 

8.  How  many  minutes  are  there  in  37°  30'  ? 

9.  How  many  minutes  are  there  in  three-eighths  of  a  right  angle? 
10.  How  many  degrees  are  there  in  three-fifths  of  a  right  angle  ? 


40  OBSERVATIONAL   GEOMETRY 

11.  How  many  degrees  are  there  in  five-sixths  of  a  right  angle? 

12.  What  part  of  a  right  angle  is  18°  ? 

13.  What  part  is  60°  ? 

14.  What  part  is  72°  ? 

15.  What  part  is  80°  ? 

16.  What  part  is  22°  30'? 

17.  How  many  right  angles  are  there  in  120°  ? 

18.  In  108°  ? 

19.  In  135°? 

20.  In  126°? 


A  Protractor 

3.  Protractors.  A  protractor  is  an  instrument  which  is 
used  to  find  the  size  of  an  angle,  or  to  construct  an  angle  of 
any  required  size.  Protractors  are  made  of  thin  metal,  ivory, 
cardboard,  etc.,  and  in  various  shapes  of  which  the  commonest 
are  shown  in  the  annexed  pictures.  They  may  be  graduated 
to  any  degree  of  minuteness;  but  intervals  of  five  degrees 
are  fine  enough  for  our  purpose. 

If  you  have  no  protractor,  you  should  make  one  out  of 
card  or  stiff  paper,  copying  one  of  the  forms  given  here. 

The  lower  straight  edge  can  be  graduated  so  as  to  serve 
for  a  measuring  ruler. 

At  the  middle  point  of  one  edge,  BA,  of  the  protractor  is  a  notch  or 
dot,  marked  Cin  the  diagram;  this  point  is  the  vertex  of  any  angle  to 


ANGLES 


\\V\\T\TI 

135        \         125         \     115       \    105      \      C5 


r  i  r  /  r  / 


83        /      75        /        65 


i  decimetre 


A  Protractor 


which  the  protractor  is  applied,  and  CA  is  placed  directly  on  one  side  of 
the  angle.  The  other  side  of  the  angle  is  indicated  by  the  little  lines  at 
the  edge  of  the  protractor,  having  numbers  which  show  the  size  of  the 
angle  in  degrees.  This  second  side  is  seldom  drawn  completely  to  the 
point  C,  because  for  convenience  in  use  most  protractors  have  an  open 
space  in  the  interior;  but  you  will  notice  that  if  the  lines  around  the  rim 
were  prolonged,  they  would  all  meet  at  the  point  C. 

In  the  first  picture  of  the  protractor  the  angles  are  numbered  from  left 
to  rght;  but  in  the  second  picture  they  are  numbered  from  right  to  left 
according  to  the  way  in  which  angles  are  supposed  to  increase  in  size. 

B 


4.   To  measure  an  angle,  as  ACS,  with  the  aid  of  a  protractor, 

place  the  protractor  as  in  the  annexed  figure,  with  the  notch 
upon  the  vertex  C,  and  the  edge  upon  one  side  CA,  so  that  the 


OBSER  VA  TIONA L   GE OME TR  Y 


point  on  the  rim  which  indicates  zero  degrees  may  be  on  CA. 
Then  observe  the  number  of  degrees  marked  on  the  rim  of 
the  protractor  where  it  is  crossed  by  the  other  side,  CB,  of 
the  angle.  This  will  be  the  number  of  degrees  in  the  angle, 
if  the  protractor  is  graduated  from  right  to  left;  but  if  from 
left  to  right,  the  number  on  the  rim  must  be  subtracted  from 
1 80°  to  show  the  size  of  the  angle. 

Estimate  by  the  eye  the  size  of  the  following  angles,  and 
then  test  your  answers  with  a  protractor. 


21-23 


24-29 


3°-35 


36-43 


44-51 


5.  To  construct  an  angle  of  required  size  with  the  aid  of  a 
protractor.  Suppose  you  wish  to  construct  an  angle  of  140°. 
Draw  a  straight  line  CA  of  any  convenient  length.  Place  the 
protractor  with  its  notch  at  C  and  the  edge  along  CA.  Then 
find  on  the  rim  of  the  protractor  the  mark  which  indicates  the 


ANGLES 


43 


55-58 


59-61 


62-65 


66-^0 


angle  140°.  Dot  the  paper  at  that  point,  remove  the  pro- 
tractor, and  draw  the  line  CB  through  the  dot.  ACB  will  be 
the  required  angle. 


Construct  the  following  angles  with  the  aid  of  a  protractor :  — 


71.  60°. 

72.  1 60°. 

73.  45°- 

74.  80°. 


75.  155°. 

76.  170°. 

77.  25°. 


78.  85°. 

79.  105°. 

80.  5°. 


Construct  angles  equal  to  the  following  with  the  aid  of  a 
protractor:  — 


44 


OBSERVATIONAL    GEOMETRY 


Construct  the  following  angles,  ruling  the  lines  but  other- 
wise aided  by  the  eye  alone :  then  test  your  angles  with  a 
protractor :  — 


95. 

96.  50 

97.  130°. 


90°. 

° 


98.  20°. 

99.  100°. 
100.    85°. 


91.  30°. 

92.  120°. 

93.  45°. 

94.  1350. 

101.  40°  and   140°,  to  have  their  vertices  at  the  same  point  and  one   side  in 
common. 

102.  130°  and  50°. 

103.  90°  and  90°. 

104.  60°,  90°.  [20°,  90°,  to  have  theii  vertices  at  the  same  point. 

105.  45°,  135°,  80°,  100°. 


CHAPTER    V 

CONSTRUCTIONS    OF    SOME    PLANE    FIGURES 

I.    To  construct  a  triangle  when  you  know  the  length  of 
one  side  and  the  size  of  the  angles  at  the  ends  of  that  side. 

Suppose  the  side  to  be  3  cm.  long  and  the  angles  at  the 
ends  of  the  side  to  be  70°   and   50°. 

C 


Draw  the  line  AB  3  cm.  long. 

At  A  draw  a  line  making  with  AB  an  angle  of  70°,  and  at 
B  a  line  making  with  AB  an  angle  of  50°,  so  that  the  two 
lines  may  meet  at  C. 

Then  ACB  will  be  the  required  triangle. 

Construct  triangles  having  the  following  sides  and  angles  :  — 

1.  Side  5  cm  ;  angles    60°  and  60°. 


2.     il 

5  cm.  ; 

90°    "    45°. 

o. 

3  cm-  ; 

70°    "    70°. 

4.     « 

4  cm.; 

100°    "    30°. 

5.     " 

3  cm-  > 

100°    "    50°. 

6.     " 

2  inches 

;   angles  60°  and  60°. 

7.     " 

3      " 

"      30°    "     45°- 

8.     " 

2        " 

«       4S°    f<     45°. 

10.       "        2        "  "         70°     "      50° 


46 


OBSERVATIONAL   GEOMETRY 


2.  The  triangle  in  which  the  two  angles  are  each  60° 
should  be  carefully  observed.  If  you  measure  the  third  angle, 
you  will  find  that  it  is  60° ;  and  if  you  measure  the  two  sides 
of  this  angle,  you  will  find  that  they  are  each  of  the  same 
length  as  the  first  side.  This  triangle,  therefore,  is  both  equi- 
angular (that  is,  "  having  all  three  angles  equal")  and  equi- 
lateral (that  is,  "  having  all  three  sides  equal"). 

If  you  wish  to  construct  an  equilateral  triangle  with  a  side,  say  5  cm. 
long,  you  can  draw  a  line  5  cm.  long,  and  at  each  end  make  an  angle  of 
60°,  prolonging  the  lines  until  they  meet.  The  resulting  triangle  will  be 
both  equilateral  and  equiangular. 


An  Equilateral  and  Equiangular  Triangle 

3.  The  sum  of  the  three  angles  of  any  triangle  is  180°,  which 
is  equal  to  two  right  angles.  You  can  test  this  by  an  experi- 
ment. A 


Draw  a  triangle  ABC of  any  shape  and  size,  and  draw  a  perpendicular 
AP  upon  one  of  the  longer  sides,  BC,  thus  forming  two  right  angles,  APB 


CONSTRUCTIONS  OF  SOME  PLANE  FIGURES        47 

and  A  PC.  Cut  out  the  triangle  from  the  paper,  and  fold  over  the  three 
vertices  upon  P.  You  will  see  that  the  three  angles  of  the  triangle  ex- 
actly cover  the  two  right  angles  beneath. 

If,  therefore,  you  know  the  size  of  two  angles  of  a  triangle, 
you  can  find  the  third  angle  by  subtracting  their  sum  from 
180°. 


Thus  if  two  angles  of  a  triangle  are  20°  and  no0,  their  sum  is  130°, 
which  leaves  50°  for  the  third  angle. 

Find  the  number  of  degrees  in  the  third  angle  of  the  fol- 
lowing triangles : 


11.  A  —  20°,  B  =    40°. 

12.  A  =  80°,  B  =    60°. 

13.  A  -  30°,  B  =  130°. 

14.  A  =  45°,  B  =    90°. 

15.  A  =  70°,  B  =    70°. 


16.  A  +  B  =  100°. 

17.  A  +  B  =  140°. 

18.  A  +  B  =    10°. 


19.  A 

20.  A 


B  = 


95 


4.    Besides  the  equilateral  triangle,  two  others  need  special 
notice, —  the  isosceles  and  the  right  triangle. 

In  an  isosceles  triangle  there  are  always  two  equal  angles, 
which  lie  opposite  the  equal  sides,  at  the  ends  of  the  base. 
The  third  angle  is  called  the  vertex  angle. 


Thus  in  the  triangle  ABC,  in  which  CA  and  CB  aie  equal,  the  angles  A  and 
B  are  equal  to  each  other;  and  C  is  the  vertex  angle. 


48 


OBSERVATIONAL   GEOMETRY 


Also,  if  you  know  that  two  angles  of  a  triangle  are  equal, 
you  can  infer  that  two  sides  are  equal,  and  the  triangle  is 
isosceles. 

Thus  in  the  triangle  DEF,  if  the  angles  D  and  E  are  known  to  be  equal,  the 
sides  FD  and  FE  are  also  equal. 


If,  therefore,  you  are  told  the  size  of  one  angle  of  an  isos- 
celes triangle,  you  need  only  to  know  whether  this  angle  is 
the  vertex  angle  or  one  of  the  equal  ones,  in  order  to  find 
the  sizes  of  the  other  two. 

Suppose,  for  example,  that  the  vertex  angle  of  an  isosceles  triangle  is  40°. 
Subtracting  40°  from  180°,  you  would  have  140°  left  for  the  sum  of  the  other 
two  angles ;  and  as  they  are  equal,  each  must  be  70°. 


If  one  of  the  equal  angles  of  an  isosceles  triangle  is  40°, 
another  angle  is  also  40° ;  these  two  together  contain  80°, 
which  leaves  100°  for  the  vertex  angle. 


CONSTRUCTIONS   OF  SOME  PLANE  FIGURES        49 

Find  the  size  of  each  angle  of  the  following  triangles,  know- 
ing that  the  triangles  are  isosceles  and  that  the  given  angle 
is  the  vertex  angle  :  — • 

21.  20°.  25.     70°.  28.  140°. 

22.  40°.  26.    45°.  29.    85°. 

23.  150°.  27.    90°.  30.     15°. 

24.  80°. 

Find  the  size  of  each  angle  of  the  following  triangles,  know- 
ing that  the  triangles  are  isosceles  and  that  the  given  angle  is 
one  of  the  equal  angles:  — 

31.  30°.  35.  15°.  38.  75°. 

32.  70°.  36.  50°.  39.  10°. 

33.  25°.  37.  35°.  40.  85°. 

34.  80°. 

5.    A  right  triangle  has  one  very  important  property  which 
we  will  now  investigate. 


41.  Draw  a  right  angle  with  sides  3  cm.  and  4  cm.  (or  f  in.  and  I  in.)  long,  and 

complete  a  right  triangle  by  drawing  the  hypotenuse. 

42.  Upon  each  side  of  the  triangle  draw  a  square. 

43.  Divide  each  square  into  smaller  ones  having  sides  I  cm.  (or  \  in.)  long,  and 

count  the  squares. 

44.  How  does  the  number  of  squares  formed  on  the  hypotenuse  compare  with 

the  sum  of  the  squares  on  the  other  two  sides  ? 

45.  Do  the  same  as  you  have  done  with  the  previous  triangle,  making  the  edges 

of  the  right  angle  5  cm.  and  12  cm.  (or  i^  in.  and  3  in.)  long. 


50  OBSERVATIONAL   GEOMETRY 

The  connection  between  the  area  of  the  square  on  the  hy- 
potenuse and  the  sum  of  the  areas  of  the  squares  on  the 
other  two  sides  is  the  same  in  any  right  triangle  as  in  the 
case  of  the  two  which  you  have  just  drawn,  and  is  expressed 
as  follows:  The  square  on  the  hypotenuse  of  a  right  tri- 
angle is  equal  to  the  sum  of  the  squares  on  the  other  two 
sides. 

To  construct  a  square,  therefore,  whose  area  shall  be  equal 
to  the  sum  of  the  areas  of  two  other  squares,  you  have  only  to 
draw  a  right  triangle  with  the  sides  of  the  right  angle  equal 
to  sides  of  the  given  squares,  and  then  draw  a  square  on  the 
hypotenuse;  this  will  be  the  required  square. 


46.  Draw  a  square  whose  area  shall  be 
equal  to  the  sum  of  the  areas  of 
the  squares  A'  and  S. 


47.  Draw  a  square  whose  area 
shall  be  equal  to  the 
sum  of  the  areas  of  the 
squares  P  and  Q. 


48.  The  annexed  figure  is  com- 
posed of  two  squares.  Copy 
the  figure  on  paper,  but  draw 
each  line  twice  as  long  as  in 
the  diagram.  Then  draw  a 
line  between  two  of  the  cor- 
ners, which  shall  be  the  side 
of  a  square  having  the  same 
area  as  your  diagram. 


CONSTRUCTIONS   OF  SOME  PLANE  FIGURES 
49.  Draw  a  square  whose  area  shall  be  twice  that  of  the  square  T. 


50.  A  man  owns  two  pieces  of  land,  one  6  rods  square  and  the  other  8  rods 
square,  which  he  is  to  exchange  for  a  single  piece  of  land  also  square- 
shaped  and  of  as  great  an  area  as  the  other  two  together.  What  is  the 
length  of  a  fence  which  will  enclose  his  new  land? 

6.    To  draw  a  straight  line  through  a  given  point  parallel 
to  a  given  straight  line. 


Using  a  Ruler  and  Square. 

(a)   With  the  aid   of   a  ruler    and  square.     Suppose   you 
wish  to  draw  through  the  point  P  a  line  parallel  to  AB, 


OBSER  VA  TIONA  L   GEOMETR  Y 


Place  the  ruler  and  square  so  that  the  edge  of  the  ruler  may  be  close 
to  P,  and  the  square  may  have  one  edge  close  to  AB  and  the  other  against 
the  ruler.  Then  slide  the  square  along  the  ruler  until  it  reaches  P. 
Along  the  edge  draw  PX,  which  will  be  the  required  line  parallel  to  AB. 

(b)    With  the  aid  of  a  "  parallel  ruler." 


Using  a  Parallel  Ruler 


CONSTRUCTIONS   OF  SOME  PLANE  FIGURES        53 

This  instrument  consists  of  two  rulers  fastened  together  with  two  strips 
of  metal  swinging  on  pivots  at  each  end.  The  distances  between  the 
pivots,  measured  along  the  metallic  strips,  are  equal ;  and  the  distances 
between  the  pivots,  measured  along  the  rulers,  are  also  equal.  Thus  the 
pivots  are  the  vertices  of  a  parallelogram  whether  the  ruler  is  open  or 
shut.  All  four  edges  of  the  ruler  are  made  parallel  to  each  other  so  that 
parallel  lines  may  be  ruled  along  any  of  them. 


To  draw  a  straight  line  through  P  parallel  to  AB,  place  the  ruler  with 
one  edge  close  to  AB,  and  hold  that  half  of  the  instrument  firmly  in  its 
place.  Swing  the  other  half  on  the  pivots  until  its  edge  reaches  P; 
then  draw  XY  along  that  edge  through  P  •  this  will  be  the  required  line 
parallel  to  AB. 


7.  To  construct  a  parallelogram  when  you  know  the 
lengths  of  two  sides  which  meet  and  the  size  of  the  angle 
between  them. 


B 


Suppose  the  sides  to  be  4  cm.  and  3  cm.  long,  and  the  angle  to  be  60°. 
Draw  the  line  AB,  4  cm.  long. 

At  A  draw  AC,  3  cm.  long,  making  with  AB  an  angle  of  60°. 
Prolong  AB  through  B  some  convenient  length  to  X. 
Draw  BD  of  the  same  length  as  AC  and  making  the  angle  XBD  of 
the  same  size  as  the  angle  A. 
Draw  the  straight  line  CD. 

Then  ABCD  will  be  the  required  parallelogram. 
BD  could  also  be  drawn  with  che  aid  of  a  square  or  a  parallel  ruler. 


54  OBSERVATIONAL   GEOMETRY 

Construct  parallelograms  having  the  following  sides  and 
angles;   and  then  tell  the  kind  of  parallelogram  each  is:  — 

51.  Sides,  5  cm.  and  2  cm. ;  angle  45°. 

52.  "  5  cm.  "    5   "           •"  60°. 

53.  "  4  cm.  "    3  "            "  90°. 

54.  "  3  cm.  "    3  "            "  90°. 

55.  "  2  inches  and  3  inches ;  angle    50°. 

56.  "  2      "        "2       "  "      120°. 

57.  "  2       "        "2       "  "        90°. 

58.  "  2                 M    I  inch  "        90°. 

The  sum  of  the  three  angles  of  any  triangle  is  two  right  angles,  or  180°. 
The  square  on  the  hypotenuse  of  a  right  triangle  is  equal  to  the  sum  of 
the  squares  on  the  other  two  sides. 


CHAPTER    VI 


Cathedral  of  the  Incarnation,  Garden  City,  N.  Y. 
A    TRUNCATED    PRISM 

OBSERVE  the  buttresses  on  the  church  in  the  picture.  They 
are  not  prisms  such  as  we  have  been  studying,  for  their  upper 
surfaces  are  inclined  to  their  bases,  but  they  are  what  are 
called  truncated  prisms,  truncated  meaning  "lopped  off."  A 


56  OBSERVATIONAL    GEOMETRY 

truncated  prism  is  one  from  which  a  part  has  been  cut  off 
by  a  plane  inclined  to  the  base. 

We  will  now  make  a  model  of  a  truncated  square  prism 
or  cube. 


A    TRUNCATED  PRISM  57 

f  he  diagram  will  need  paper  19  cm.  X  17  cm.  (or  7^  in.  X  6^  in.). 
The  construction  can  be  seen  from  the  special  figure. 
A,  B,  C,  and  D  are  squares  with  edges  5  cm.  (or  2  in.)  long. 
L  is  a  rectangle  with  the  shorter  edges  2  cm.  5  mm.  (or  I  in.)  long. 
From  two  corners  of  A  lines  X  are  drawn  to  the  middle  points  of  the  outer 
edges  of  two  adjoining  squares. 

E  is  a  rectangle  with  the  longer  edges  equal  to  X. 

The  base  of  the  truncated  prism  is  the  base  of  the  prism 
of  which  it  was  a  part. 

The  face  formed  by  the  cutting  plane  is  called  the  inclined 
section. 

The  other  faces  are  called  lateral  or  side  faces. 

1.  What  are  the  shapes  of  the  lateral  faces  of  this  figure? 

2.  If  you  were  to  place  the  figure  upon  one  of  its  lateral  faces  as  a  base,  what 

should  you  then  call  the  figure  ? 

3.  How  does  it  happen  that  this  figure  has  different  names  according  to  its 

position  ? 

4.  Supposing  the  original  figure  to  have  been  a  cube,  can  you  see  what  the 

shape  of  the  part  cut  off  must  have  been  ? 

5.  Can  you  place  two  of  these  truncated  prisms  together  so  as  to  form  a 

rectangular  parallelepiped  ? 

6.  What  would  be  the  volume  of  that  parallelepiped  ? 
7-  What,  then,  is  the  volume  of  your  truncated  prism  ? 


CHAPTER   VII 


The  Pyramids  of  Egypt 
A    PYRAMID 

I.  IN  the  above  picture  you  see  a  very  ancient  geometric 
form,  supposed  to  have  been  invented  by  the  Egyptians. 
This  is  the  pyramid  (pyr'-a-mid),  a  word  whose  original 
meaning  is  uncertain,  though  some  writers  trace  a  connection 
between  it  and  a  word  meaning  "  fire." 

A  pyramid  has  all  its  faces,  except  one,  triangles  which 
meet  at  a  point  called  the  apex  (a'-pex),  a  word  meaning 
"  a  top  or  summit."  The  other  face,  which  may  have  any 
number  of  edges,  is  called  the  base ;  and  a  pyramid  takes  a 


A   PYRAMID 


59 


name  —  square,  triangular,  etc.  —  according  to  the  shape  of 
its  base. 

We  will  now  make  a  model  of  a  pyramid  having  a  square 
base. 


6o 


OBSEX  VA  TLONA  L    GE  OME  TR  Y 


2.  The  diagram  will  need  paper  16  cm.  5  mm.  X  16  cm.  or 
The  construction  can  be  seen  from  the  special  figure. 

First  draw  a  square  with  edges  5  cm.  (or  2  in.)  long,  and  find  their  middle 
points,  M,  AT,  J?,  and  S. 

Then  draw  outward  from  the  edges  the  perpendicular  lines  MX,  NY.  RZ, 
and  SWt  each  5  cm.  6  mm.  (or  z\  in.)  long. 

Then  draw  XA,  XB,  YB>  etc.,  to  the  corners  of  the  square. 
I 

3.  1.  How  many  faces  has  this  pyramid  ? 

2.  How  many  edges  ? 

3.  How  many  corners  ? 

4.  How  many  angles  have  the  faces  altogether? 

5.  How  many  of  the  edges  are  perpendicular  to  other  edges-? 

6.  What  is  the  greatest  number  of  edges  perpendicular  to  any  one  edge? 

4.  DiedraJ  Angles  You  have  seen  how  the  edges  of 
faces  may  make  angles  with  other  edges.  You  will  now 
notice  that  the  faces  themselves  may  make  angles  with  other 
faces,  and  in  fact  always  do  if  they  are  extended  far  enough, 
unless  they  are  parallel.  But  notice  that  instead  of  meeting 
at  a  point  as  two  edges  do,  two  faces  meet  in  a  straight  line. 
An  angle  formed  by  two  faces  is  called  a 
diedral  (di-e'-dral)  angle. 

Diedral  means  "  having  two  sides." 

Diedral  angles,  like  line  angles,  may  be 
acute,  right,  or  obtuse. 

7.  How   many   diedral   angles    does   the    base  of  the 

square  pyramid  form  with  the  other  faces? 

8.  Do  these  angles  seem  to  you  to  be  acute,  right,  or 

obtuse  ? 

9.  How  many  diedral  angles  are  made  by  the  triangu- 

lar faces  among  themselves? 
10.  Of  what  kind  do  these  angles  seem  to  you  to  be  ? 

Diedral  angles  may  be  measured  with  the  aid  of  a  stiff 
rectangular  card,  five  or  six  inches  long  and  two  inches  wide, 
folded  so  that  the  shorter  edges  may  be  exactly  together. 

The  card  is  placed  with  its  folded  edge  against  the  edge 
of  the  angle  to  be  measured,  the  halves  of  the  card  lying 


A  PYRAMID  61 


Measuring  a  Diedral  Angle 

closely  upon  the  faces  of  the  angle.  The  protractor  is  then 
placed  with  the  notch  at  one  end  of  the  folded  edge  of  the 
card,  and  the  angle  between  the  two  diverging  edges  of  the 
card  is  measured:  this  angle  is  equal  to  the  diedral  angle  of 
the  faces. 

You  should  now  practise  measuring  the  diedral  angles 
between  the  faces  of  the  other  figures  you  have  made. 

5.  Area  of  a  Triangle.  The  surface  of  your  pyramid  is 
composed  of  a  square  base  and  four  triangles.  You  know 
already  how  to  find  the  area  of  the  base;  and  we  will  now 
examine  the  other  faces. 

These  faces  are  triangles.  The  area  of  a  triangle  is  equal 
to  the  length  of  any  one  of  its  sides  multiplied  by  one-half 
the  perpendicular  distance  of  that  side  from  the  opposite 
vertex. 


62 


OBSER  VA  TIONAL   GEOME TR  Y 


We  will  first  calculate  the  area  of  one  of  the  triangles  from 
the  diagram  which  you  used  in  constructing  the  pyramid, 
and  then  test  the  answer  by  measurement. 


M 

In  the  triangle  AXB,  what  length  is  represented  by  AB  ? 

What  length  is  represented  by  XM? 

How  much  is  one-half  the  length  of  XM ' ? 

Multiplying  one-half  of  XMby  AB  gives  what  for  the  area  of  the  triangle  ? 

Now  let  us  test  this  by  measurement. 

Construct  on  paper  a  triangle  of  the  exact  size  of  one  of  the  faces :  — 
Draw  AB,  5  cm.  (or  2  in.)  long,  and  find  the  middle  point  M. 


At  ^/draw  the  perpendicular  AfX,  5  cm.  6  mm.  (or  2!  in.)  long,  and 
then  draw  XA  and  XB, 

Cut  the  triangle  carefully  out  from  the  paper. 

Fold  over  the  upper  part  so  that  X  may  come  exactly  upon  M,  and 
make  the  crease  represented  by  EPF.  Cut  off  the  part  EXF  along  the 
crease,  and  cut  this  again  into  two  parts  along  the  line  XP '.  Then  match 
these  two  parts  upon  the  rest  of  the  triangle  as  shown  in  the  figure. 

You  have  now  turned  the  triangle  into  a  rectangle,  which 
you  can  paste  together  with  a  strip  of  paper  on  the  back. 


A    PYRAMID  63 

11.  What  is  the  length  of  this  rectangle  ? 

12.  What  is  the  width? 

13.  What  is  the  area  ? 

11.  Does  this  result  agree  with  the  answer  you  obtained  by  calculation  from  the 
diagram?  If  not,  see  if  you  can  find  where  you  have  made  a  mistake. 
The  area  is  14  sq.  centimetres,  or,  if  you  have  used  English  measurements, 
2\  sq.  inches. 

15.  What  is  the  area  of  the  four  triangles  together  ? 

16.  What  is  the  area  of  the  entire  surface  of  the  pyramid? 

6.   The  volume  of  a  pyramid  is  equal  to  one-third  of  its 
height  multiplied  by  the  area  of  its  base. 

We  will  test  this  by  trying  an  experiment  with  the  pyramid 
and  cube. 


Testing  the  Height  of  a  Pyramid 

First,  place  the  base  of  the  pyramid  against  the  base  of  the  cube,  and 
see  that  they  have  the  same  area.  Then  set  the  two  figures  on  a  hori- 
zontal plane  at  a  little  distance  apart,  and  rest  a  ruler  across  the  top  of 
the  cuba  and  apex  of  the  pyramid :  see  if  the  ruler  is  horizontal  ;  you 
will  find  it  to  be  almost  exacf.y  so  if  you  have  made  the  two  figures 
correctly.  So  the  heights  of  the  cube  and  pyramid  are  equal,  as  well  as 
the  bases.  Now  the  volume  of  the  cube  is  equal  to  the  area  of  its  base 


64  OBSERVATIONAL    GEOMETRY 

multiplied  by  the  whole  of  its  altitude ;  so  if  the  volume  of  the  pyramid 
is  one-third  of  this,  the  pyramid  ought  to  hold  only  one-third  as  much  as 
the  cube. 

Make  a  new  pyramid,  therefore,  so  as  to  save  the  other,  but  before 
pasting  the  last  edge  cut  off  the  square  base.  Then  take  the  cube  which 
you  keep  for  measuring,  and  using  sand  or  water  as  you  did  before,  see  if 
the  pyramid  has  to  be  filled  three  times  in  order  to  fill  the  cube  once. 

17.  How  many  cubic  centimetres,  therefore,  are  there  in  the  volume  of  your 

pyramid? 

18.  How  many  fillings  of  your  pyramid  would  make  one  of  the  parallelepiped 

described  on  page  17  ? 

19.  If  the  edge  of  the  base  of  your  pyramid  were  twice  as  long  as  it  is,  what 

would  be  the  volume  ? 

20.  If  the  base  were  of  the  same  size  as  it  is,  but  the  height  were  twice  as  great, 

what  would  be  the  volume  ? 

21.  What  would  be  the  volume  of  a  pyramid  with  a  height  6  inches  and  a  base 

containing  9  square  inches  ? 

22.  If  a  pyramid  and  a  cube  have  equal  bases  containing  16  square  inches,  what 

must  be  the  height  of  the  pyramid  so  that  the  volumes  of  the  two  figures 
may  be  equal  ? 

23.  How  many  pyramids,  each  with  a  height  of  3  cm.  and  a  base  area  of  16  sq. 

cm.,  could   be  filled  from  the  contents  of  a  rectangular  parallelepiped 
4  X  6  X  8cm.? 

The  area  of  a  triangle  is  one-half  the  product  of  its  base  and  altitude. 

.  .       ,       base  x  altitude      base         ....    ,        ,           altitude 
Area  triangle  =  —     — —  = x  altitude  =  base  x — 

The  volume  of  a  pyramid  is  one-third  the  product  of  the  area  of  its 
base  and  altitude. 

.,      base  x  altitude      base  altitude 

Volume  pyramid  —  -  = x  altitude  =  base  x 

3  3  J 


CHAPTER   VIII 


A    TRIANGULAR    PYRAMID 

I.  THE  diagram  will  need  paper  IQ  cm.  X  n  cm.,  or  4  in.  X  42  in.  ABC\s, 
an  equilateral  triangle,  each  edge  being  I  dcm.  long.  The  middle  points  of  the 
edges,  £>,  E,  and  F,  are  joined,  and  the  triangle  is  thus  divided  into  four  smaller 
equilateral  triangles  having  edges  5  cm.  long. 

In  English  measurements,  the  edges  of  ABC  may  be  4  inches  long,  thus 
making  the  edges  of  the  smaller  triangles  2  inches  long 


66 


OBSEX  VA  TIONA  L   GEOME  TR  Y 


2.  1.  How  many  faces  has  this  figure  ?     What  is  their  shape  ? 

2.  How  many  edges  ?     What  is  their  length  ? 

3.  How  many  corners  ? 

4.  How  many  line  angles  ?     What  is  their  size  ? 

5.  How  many  diedral  angles  ?     What  is  their  size? 

6.  This  figure  is  called  a  pyramid:  why? 

7.  It  is  also  called  a  triangular  pyramid :  why  ? 

8.  Has  it  more  than  one  face  which  can  be  called  its  base  ? 

9.  Has  a  quadrangular  pyramid  more  than  one  such  face  ? 
10.  Can  you  explain  the  difference  between  these  two  cases  ? 


3.  Solid  Angles.  We  have  seen  that  when  two  edges 
meet,  or  would  meet  if  prolonged,  they  form  a  line  angle ; 
and  when  two  faces  meet,  or  would  meet  if  extended,  they 
form  a  diedral  angle.  Now  when  three  or  more  faces  meet 
at  a  point  and  enclose  all  the  space  around  the  point,  they 
make  what  is  called  a  solid  angle. 


A  Solid  Angle 


If  you  observe  figures  carefully  you  will  see  that  it  takes  at  least  three 
faces  to  form  a  solid  angle ;  for  .two  faces  would  leave  an  open  space. 
But  there  may  be  as  many  faces  as  you  please  more  than  three  ;  though 
if  you  try  to  make  a  solid  angle  by  joining  pieces  of  paper  you  will  find 
that  the  sum  of  the  angles  formed  by  the  edges  must  not  be  so  great  as 
360°  or  4  right  angles.  If  the  sum  were  equal  to  360°,  the  pieces  of 
paper  would  lie  flat  and  form  a  plane. 

Notice  that  a  solid  angle  has  an  open  space  in  front  of  the  point  If 
this  space  were  closed  by  a  plane  cutting  the  other  faces,  the  resulting 
figure  would  be  a  pyramid. 


A    TRIANGULAR  PYRAMID 


A  Solid  Angle 

If  a  solid  angle  is  formed  by  three  faces,  it  is  called  a 
triedral  (tri-e'-dral)  angle,  which  means  "  having  three  faces." 

If  it  is  formed  by  four  or  more  faces,  it  is  called  a  polyedral 
(pol-y-e'-dral)  angle,  which  means  "  having  many  faces." 

11.  What  is  the  difference  between  a  triedral  angle  and  a  triangular  pyramid  ? 

12.  How  many  solid  angles  has  a  triangular  pyramid? 

13.  How  many  has  a  cube  ?     What  is  the  sum  of  the  line  angles  which  form 

each  solid  angle  ? 

14.  How  many  solid  angles  has  a  quadrangular  pyramid? 

15.  Is  there  a  solid  angle  at  each  corner  of  a  figure  which  is  entirely  enclosed 

by  planes  ? 

16.  In  the  triangular  pyramid  is  the  number  of  solid  angles  equal  to  the  number 

of  faces  ? 

17.  Is  this  true  of  the  cube  ? 

18.  Of  the  triangular  prism  ? 

19.  Of  the  quadrangular  pyramid? 


CHAPTER   IX 


A    PENTAGONAL    PYRAMID 

I.  The  diagram  will  need  paper  15X15  cm.,  or  6  X  6  inches. 
Draw  AB,  3  cm.  long. 
At  A  draw  AE,  3  cm.  long,  and  making  the  angle  BAE  108°. 


A   PENTAGONAL   PYRAMID  69 

At  B  draw  BC,  3  cm.  long,  and  making  the  angle  ABC  108°. 
At  E  draw  ED,  3  cm.  long,  and  making  the  angle  AED  108°. 
Draw  the  line  DC  completing  the  interior  part  of  the  diagram. 
Prolong  the  lines  AB,  BC,  etc.,  in  both  directions  until  they  form  the  five, 
pointed  star  FGHIJ. 

With  English  measurements,  i  inch  will  be  a  convenient  length  for  AB. 

2.  This   figure    is    called  a  pentagonal  (pen-tag'-o-nal) 
pyramid.     Its  base  is  a  pentagon  (pen'-ta-gon),  which  means 
a  face  having  five  corners.     It  also  has  five  edges ;   for  every 
face  has  as  many  edges  as  it  has  corners. 

3.  Examine  the  completed  model,  make  measurements, 
and  write  out  your  answers  to  the  following :  — 

1.  The  number  of  faces. 

2.  The  number  of  edges. 

3.  The  number  of  corners. 

4.  The  shapes  of  the  faces  and  the  number  of  each  shape. 

5.  The  lengths  of  the  edges  and  the  number  of  each  length. 

6.  The  number  of  face  angles. 

7.  The  sizes  of  the  face  angles  and  the  number  of  each  size. 

8.  The  number  of  diedral  angles. 

9.  The  sizes  of  the  diedral  angles  and  the  number  of  each  size 

10.  The  number  of  solid  angles. 

11.  The  number  of  faces  which  form  each  solid  angle. 


CHAPTER   X 


A    HEXAGONAL    PYRAMID 


I.  The  diagram  will  require  paper  20  X  20  cm.  or  8  X  8  inches. 

Construct  the  equilateral  triangle  XYZ,  each  side  9  cm.  (or  4^  in.)  long. 

Divide  each  side  into  equal  parts,  each  3  cm.  (or  i|  in.)  long,  the  points  of 
division  being  A,  B,  C,  D,  E,  and  F. 

Draw  AB,  CD,  and  EF,  thus  completing  the  interior  part  of  the  diagram 
ABCDEF,  which  has  six  dotted  sides  all  equal. 

Upon  each  of  the  six  sides  AB,  BC,  CD,  etc.,  construct  an  isosceles  triangle 
with  the  angles  at  A,  B,  C,  etc.,  each  75°,  thus  forming  the  six-pointed  star 
GHIJKL. 


A    HEXAGONAL  PYRAMID 
J 


CHAPTER  XI 

POLYGONS    AND    SYMMETRY 

I.  You  have  been  told  that  pyramids  take  their  names 
from  the  shape  of  their  bases.  Now  the  bases,  like  all  faces, 
take  their  names  as  follows  :  — 

First,  from  the  number  of  edges  or  corners,  the  number  of 
edges  being  the  same  as  the  number  of  corners. 

Secondly,  from  equality  in  the  length  of  the  edges. 

Thirdly,  from  equality  in  the  size  of  the  angles. 

Fourthly,  from  equality  in  both  edges  and  angles. 

Fifthly,  from  peculiarity  in  the  arrangement  of  the  edges 
or  angles. 

The  general  name  for  a  face  is  polygon  (pol'-y-gon),  which 
means  "  having  many  corners ;  "  but  this  name  is  usually 
applied  only  to  faces  which  have  more  than  four  corners, 
that  is,  more  than  four  edges. 


Equilateral  Polygons 

If  the  edges  of  a  face  are  all  equal  to  each  other,  it  is  called 
an  equilateral  polygon. 


POLYGONS  AND  SYMMETRY 


73 


If  the  angles  of  a  face  are  all   equal  to  each  other,  it  is 
called  an  equiangular  polygon. 


Equiangular  Polygons 

If  a  face  is  both  equilateral  and  equiangular,  it  is  called  a 
regular  polygon. 


Regular  Polygons 

2.   A  polygon  is  symmetrical  with  respect  to  a  straight  line 

when  this  line  divides  it  into  two  parts  such  that,  if  the  figure 


Symmetrical  Polygons 

be  revolved  on  the  line,  the  two  parts  will  exchange  places, 
each  exactly  covering  the  space  formerly  occupied  by  the 
other.  The  straight  line  is  called  the  axis  of  symmetry. 


74 


OBSERVATIONAL   GEOMETRY 


You  can  test  this  by  an  experiment.  First  construct  a  symmetrical 
polygon  as  follows :  Draw  a  square  ABCD  with  edges  4  cm.  (or  2  in.) 
long. 

A  E  D 


M 


Draw  EF  connecting  the  middle  points  of  two  opposite  edges  AB 
and  DC.  Divide  each  edge  into  four  equal  parts. 

Draw  PL  and  MN  connecting  the  points  of  division  nearest  D  and  C. 

Draw  EP  and  EN. 

A  symmetrical  polygon  LMNEP  will  thus  be  formed,  of  which  EF\$> 
the  axis.  Cut  out  this  polygon,  using  a  ruler  and  knife  so  as  to  preserve 
the  edges  of  the  gap  left  in  the  paper.  Then  turn  the  polygon  over,  and 
replace  it  in  the  paper  in  the  reversed  position.  You  will  see  that  the 
ends  of  the  axis  EFare  in  their  former  position ;  but  N  and  />,  and  M 
and  Z,  have  exchanged  places;  thus  all  points  in  the  polygon,  except 
those  in  the  axis,  have  exchanged  places  with  other  points. 

Draw  the  following  figures,  all  of  which  are  symmetrical 
with  respect  to  a  line,  and  then  draw  their  axes :  — 

1.  An  isosceles  triangle. 

2.  A  straight  line. 

3.  An  angle  with  equal  sides. 

4.  An  equilateral  triangle  (three  axes). 

5.  A  square  (four  axes). 

6.  A  straight  line  met  at  its  middle  point  by  two  equal  straight  lines  so  as  to 

form  three  angles  each  60°. 

7.  A  rectangle  (two  axes). 

8.  A  parallelogram  with  equal  angles. 

9.  A  rhombus  (two  axes). 

10.  A  trapezoid  with  two  equal  sides. 

Two  figures  when  considered  together  may  be  symmetrical 
with  respect  to  a  line. 


POLYGONS  AND  SYMMETRY  75 

For  example,  you  can  draw  a  polygon  in  ink,  and  before  the  ink  is  dry  you 
can  fold  the  paper  so  as  to  make  an  impression  on  some  other  part.  Then  the 
polygon  and  its  impression  will  be  symmetrical  with  respect  to  the  crease  in  the 
paper  which  will  represent  the  axis. 

3.  A  figure  is  said  to  be  symmetrical  with  reference  to  a  point 
when,  being  turned  half-way  round  on  this  point  as  a  pivot, 
it  exactly  covers  every  part  of  the  surface  which  it  occupied 
in  its  former  position.  The  pivot  point  is  called  the  centre 
of  symmetry.  In  this  case  a  figure  keeps  in  its  own  plane 
throughout  its  revolution  ;  whereas,  when  it  revolves  on  an 
axis,  it  leaves  the  plane  at  once  and  returns  to  it  only  when 
the  revolution  is  completed. 


You  can  test  this  by  an  experiment. 

Draw  a  parallelogram  ABCD,  and  connect  its  opposite  corners  with 
straight  lines ;  the  point  P  where  these  lines  cross  will  be  the  point  of 
symmetry.  Cut  out  the  figure  with  a  knife,  preserving  the  edges  care- 
fully. Return  the  figure  to  its  place,  and  stick  a  pin  through  the  pivot 
point.  Then  revolve  the  figure  about  P  until  the  gap  in  the  paper  is 
filled  again.  You  will  see  that  every  point  except  the  pivot  has  moved, 
each  exchanging  places  with  another  which  was  equally  distant  from  the 
pivot ;  thus  A  changes  places  with  D,  and  B  with  C. 

Draw  the  following  figures,  all  of  which  are  symmetrical 
with  respect  to  a  point,  and  indicate  the  point  in  each  case 
by  the  letter />:-— 

11.  A  straight  line. 

12.  A  square. 

13.  A  straight  line  with  two  equal  perpendicular  lines,  one  from  each  end,  ex- 

tending in  opposite  directions. 

14.  A  rhombus. 

15.  Two  unequal  straight  lines  cutting  each  other   perpendicularly  into  equal 

parts. 


76  OBSERVATIONAL   GEOMETRY 

16.  Two  equal  parallel  lines. 

17.  Two  unequal  straight  lines  cutting  each  other  into  equal  parts,  but  not 

perpendicularly. 

18.  A  rectangle. 

19.  A  straight  line  from  the  ends  of  which  extend  two  equal  lines  on  opposite 

sides  of  the  first  line  with  which  each  makes  an  angle  of  60°. 

20.  A  straight  line  at  the  ends  of  which  are  two  parallel  equal  lines  which  the 

first  line  divides  into  equal  parts. 

Two  figures  when  considered  together  may  be  symmetrical 
with  respect  to  a  point. 

Cut  out  a  polygon  of  any  shape.  Make  a  tracing  of  this  on  paper ;  then 
turn  the  polygon  half-way  around  so  that  one  edge  may  be  in  a  straight  line 
with  its  former  position,  and  make  another  tracing.  The  two  tracings  combined 
will  be  symmetrical  with  respect  to  a  point  half-way  between  the  two  nearest 
vertices. 

\ 

In  the  preceding  examples  we  have  what  is  called  twofold 
symmetry  with  reference  to  a  point.  An  equilateral  triangle 
is  an  example  of  threefold  symmetry;  in  this  case  the  figure 
when  revolved  only  one-third  of  the  way  around  the  point 
occupies  the  same  surface  it  covered  at  first,  and  after  three 
revolutions  it  returns  to  its  original  position.  In  the  same 
way  the  base  of  the  pentagonal  pyramid  of  Chapter  IX.  has 
fivefold  symmetry.  All  regular  polygons  have  a  symmetry 
of  as  many  fold  as  they  have  sides.  Furthermore,  a  figure 
may  have  symmetry  of  several  kinds ;  thus  the  base  of  the 
hexagonal  pyramid  of  Chapter  X.  has  two,  three,  and  six  fold 
symmetry. 

Of  how  many  fold  is  the  symmetry  of  the  figures  in 
Chapter  XXV.  I3,of  Part  II.,  as  follows?  — 


21.  Problem    i.  25.  Problem  16. 

22.  "          5.  26.  "       20. 

23.  "       ii.  27.  "       24. 

24.  "       14.  28.  «        25. 


POLYGONS  AND  SYMMETRY 


77 


29.  Are  your  hands  symmetrical  with  respect  to  a  point  or  to  a  line  ? 

30.  Of  which  kind  is  the  symmetry  of  flowers  such  as  the  clematis  and  narcissus  ? 

31.  In  which  kind  of  symmetry  do  leaves  usually  appear  on  a  branch  ? 

4.   The  perimeter  (pe-rim'-e-ter)  of  a  polygon  is  the  sum 
of  the  edges  which  bound  it. 

The  word  means  "  a  measuring  around." 

Calculate  the  perimeters  of  the  following  figures:  — 

32.  A  rhombus  whose  edge  is  5  cm. 

33.  A  rectangle  whose  length  is  5  cm.  and  width  3  cm. 

34.  A  square,  two  of  whose  edges  combined  are  8  cm.  long. 

35.  A  parallelogram  two  of  whose  edges  are  3  cm.  and  7  cm. 

36.  A  parallelogram  in  which  the  distance  from  one  corner  to  the  opposite 

corner,  measured  along  the  edges,  is  12  cm. 

37.  An  equilateral  triangle,  one  of  whose  edges  is  4  cm. 

38.  An  equilateral  triangle,  two  of  whose  edges  combined  are  10  cm.  long. 

39.  An  equilateral  polygon  bounded  by  eight  edges,  one  of  which  is  2  cm. 

43.  An  equilateral  polygon  bounded  by  twelve  edges,  five  of  which  combined 
are  15  cm.  long. 


5.  A  diagonal  (di-ag'-o-nal)  of  a  polygon  is  a  straight  line 
drawn  between  any  two  corners  which  are  jiot  already  con- 
nected by  an  edge. 

The  word  means  "through  the  corners."  Thus  ^C*and  AD  are  diagonals, 
but  no  diagonal  can  be  drawn  between  A  and  B,  since  these  corners  are  already 
connected  by  the  edge  AB. 


78  OBSERVATIONAL   GEOMETRY 

41.  How  many  diagonals  can  you  draw  from  any  one  corner  of  a  rectangle  ? 

42.  How  many  different  diagonals  can  you  draw  between  all  the  corners  of  a 

rectangle  ? 

43.  If  you  draw  a  diagonal  of  a  square  what  is  the  shape  of  the  parts  into  which 

you  divide  it  ? 

44.  Why  cannot  diagonals  be  drawn  in  a  triangle  ? 

45.  Draw  a  polygon  of  five  edges  and  then  draw  all  the  diagonals  you  can  from 

any  one  corner. 

46.  Into  how  many  parts  have  you  divided  the  polygon  ? 

47.  What  is  the  shape  of  the  parts  ? 

6.  Polygons  have  names  according  to  the  number  of  the 
edges.     The  following  is  a  list  which  you  may  use  for  refer- 
ence without  trying  to  commit  the  names  to  memory. 

,A  pentagon  has  five  edges ;  the  word  means  "five  corners,"  but  every  polygon 
has  as  many  edges  as  it  has  corners. 
A  hexagon  has  six  edges. 
A  heptagon,  seven  edges. 
An  octagon,  eight  edges. 
A  nonagon,  nine  edges. 
A  decagon,  ten  edges. 
An  undecagon,  eleven  edges. 
A  dodecagon,  twelve  edges. 
A  pentedecagon,  fifteen  edges. 
An  icosagon,  twenty  edges. 

7.  Distortion  of  Polygons.     A  polygon  can  have  a  count- 
less variety  of  shapes  without  changing  the  lengths  of  its 
edges. 


You  can  test  this  by  taking  small  wooden  rods  to  represent  some  poly- 
gon and  fastening  the  ends  together  with  pieces  of  tape  which  will  act 
as  hinges.  You  will  find  that  whenever  you  pulJ.  or  push  the  figure  it 
will  take  a  different  shape. 


POLYGONS  AND   SYMMETRY 


79 


Even  a  four-sided  figure  can  be  made  to  change  its  form ; 
for  a  square  can  be  turned  into  a  rhombus,  and  a  rectangle 
into  a  parallelogram. 

Triangles  are  the  only  exception.  When  a  triangle  is  once 
formed,  you  cannot  alter  its  shape  without  disturbing  the 
lengths  of  the  edges. 


Horticultural  Building  under  Construction 

Carpenters  make  an  important  use  of  these  facts  when  they  erect  the 
frames  of  buildings  or  construct  scaffolding. 

The  annexed  picture  represents  a  part  of  the  Horticultural  Building  at 
the  World's  Fair  in  Chicago,  as  it  appeared  in  the  process  of  erection. 
The  vertical  and  horizontal  beams  in  the  scaffolds  form  a  series  of 
rectangles,  which  might  collapse  under  pressure  even  if  the  fastenings 
held  firm.  But  you  will  notice  that  each  rectangle  has  two  boards  nailed 
diagonally  across,  turning  it  into  four  triangles  whicl^from  their  shape 
add  a  geometric  strength  to  that  of  the  fastenings. 

Another  common  example  is  a  gate.  This  would  be  likely  to  "  sag  " 
after  a  time ;  that  is,  it  would  change  from  a  square  or  a  rectangle  to  a 
rhombus  or  a  parallelogram  ;  but  the  cross-bar  extending  from  corner  to 


8o 


OBSERVATIONAL   GEOMETRY 


corner  turns  the  quadrilateral  into  two  triangles,  which  must  keep  their 
shape  unless  the  wood  decays  or  the  joints  become  loose. 

Let  us  see  how  many  cross  pieces  are  needed  to  hold  a 
polygon  to  its  shape.  We  have  seen  that  a  carpenter  uses 
two  for  each  rectangle  in  his  scaffold  and  only  one  in  a  gate, 


though  both  figures  are  quadrilaterals.  But  the  carpenter 
has  to  consider  the  strength  of  materials,  and  a  long  beam  is 
more  likely  to  bend  than*  a  short  one.  Otherwise  the  ques- 
tion would  simply  be:  how  many  diagonals  must  you  draw 
in  a  polygon  to  divide  it  into  triangles?  Make  experiments 
with  polygons  of  various  numbers  of  edges,  choosing  one 
corner  from  which  to  draw  all  the  diagonals  in  each  case. 
You  will  find  that  the  necessary  number  is  always  three  less 
than  the  number  of  the  edges. 


CHAPTER    XII 


Washington's  Headquarters  at  Cambridge 


A    FRUSTUM    OF    A    PYRAMID 

I.  OBSERVE  the  roof  in  the  picture  of  Washington's 
Headquarters.  The  sides  slope  upward  from  the  eaves  as 
if  they  were  to  meet  at  an  apex  and  form  the  lateral  faces  of 
a  pyramid ;  but,  instead,  they  are  cut  short  by  the  flat  top  of 
the  roof,  and  we  have  only  a  part  of  a  pyramid:  such  a  part 
is  called  a  frustum,  a  word  meaning  a  "bit"  or  "piece"  of  any- 
thing. If  a  plane  be  passed  through  a  pyramid  parallel  to  its 
base,  and  the  upper  part  (between  the  plane  and  the  apex) 
be  removed,  the  rest  of  the  figure  is  called  the  frustum  of  a 
pyramid. 


82 


OBSERVATIONAL   GEOMETRY 


We  will  now  make  a  model  of  the  frustum  of  a   square 
pyramid. 


2.   The  diagram  will  need  paper  14  cm.  X  12  cm.  (or  5!  in.  X  5  in.)- 

A  is  a  square  with  edges  5  cm.  (or  2  in.)  long. 

B,  C,  Z>,  and  E  are  equal  trapezoids ;  one  edge  of  each  is  5  cm.  (or  2  in.) 
long,  and  other  edges  are  all  2  cm.  5  mm.  (or  i  in.)  long;  the  angles  at  the  ends 
of  the  longer  edges  are  all  60°. 

/''is  a  square  with  edges  2  cm.  5  mm.  (or  i  in.)  long. 

3.  The  frustum  has  two  bases.  The  lower  base  is  the 
base  of  the  pyramid  itself,  and  may,  therefore,  have  any 
number  of  edges  and  any  shape.  The  upper  base  is  formed 
by  the  cutting  plane,  a'nd  is  an  exact  reduced  copy  of  the 
lower  base. 


A   FRUSTUM   OF  A    PYRAMID 


Similar  Polygons 

Two  such  polygons,  which  are  shaped  exactly  alike,  one 
being  a  reduced  copy  of  the  other,  are  called  similar  polygons. 

The  other  faces  of  the  frustum,  that  is,  the  lateral  faces,  are 
always  trapezoids.  They  may  or  may  not  be  equal  or 
similar  to  one  another. 

1.  What  is  the  area  of  the  lower  base  of  the  frustum  which  you  have  made? 

2.  What  is  the  area  of  the  upper  base  ? 

3.  What  is  the  shape  of  that  part  of  the  pyramid  which  is  removed  to  form  the 

frustum  ? 

4    If  a  prism  were  cut  by  a  plane  parallel  to  the  base,  what  would  be  the  shapes 
of  the  parts  into  which  the  prism  would  be  divided  ? 


CHAPTER    XIII 


The  Castle  of  Chillon 


A    TRUNCATED    PYRAMID 

I.  IF  you  examine  the  roofs  of  the  two  highest  towers  of 
the  Castle  of  Chillon,  you  will  see  that  while  they  are  parts 
of  pyramids  they  are  not  frustums,  for  the  top  of  each  tower 
is  not  a  plane  but  an  edge.  The  cutting  plane,  therefore,  is 
not  parallel  to  the  base. 

If  a  pyramid  be  cut  by  a  plane  not  parallel  to  the  base, 
and  the  part  between  this  plane  and  the  apex  be  removed, 
the  rest  of  the  figure  is  called  a  truncated  pyramid. 


A    TRUNCATED  PYRAMID  85 

In  the  present  instance  the  pyramids  are  cut  by  two  planes, 
each  extending  to  the  base,  and  the  result  is  a  form  which  is 
common  in  architecture,  being  what  is  called  "  a  hip  roof." 

We  will  now  make  a  model  of  a  truncated  pyramid  formed 
by  a  single  cutting  plane. 


2.   The  diagram  will  need  paper  16  cm.  X  14  cm.  (or  6|  in.  X  5z  in.). 
The  construction  can  be  seen  in  the  special  figure. 


86 


OBSER  VA  TIONA  L   GE  OME  TR  Y 


A  (see  figure)  is  a 
square  with  edges  6  cm. 
(or  3  in.)  long;  upon 
each  edge  an  equilateral 
triangle  is  drawn. 

B  is  a  trapezoid 
formed  by  marking  off 
on  the  edges  of  one  of 
the  triangles  distances  of 
4  cm.  (or  2  in.)  from  the 
corners  of  the  square, 
and  drawing  the  fourth 
edge  y,  which  will  be  2 
cm.  (or  i  in.)  long. 

D  and  E  are  trape- 
ziums formed  by  marking 
off  on  the  edges  of  two 
opposite  triangles  dis- 
tances of  4  cm.  and  2 
cm.  (or  2  in.  and  I  in.) 
from  the  corners  of  the 
square,  and  drawing  the  fourth  edges  x,  which  will  meet  one  edge  of  each 
triangle  perpendicularly. 

C  is  a  trapezoid  formed  by  marking  off  on  the  edges  of  the  last  triangle  dis- 
tances of  2  cm.  (or  i  in.)  from  the  corners  of  the  square,  and  drawing  the  fourth 
edge  z,  which  will  be  4  cm.  (or  2  in.)  long. 

L  is  a  trapezoid  formed  by  drawing  a  perpendicular  at  the  middle  of  z,  33 
mm.  (or  if  in.)  long,  and  drawing  the  second  edge  w  2  cm.  (or  i  in.)  long, 
parallel  to  z  and  divided  equally  by  the  perpendicular;  the  other  two  edges  can 
then  be  drawn  and  will  each  be  equal  to  x. 

3.  The  base  of  a  truncated  pyramid  is  the  base  of  the 
pyramid  itself,  and  may,  therefore,  have  any  number  of  edges 
and  any  shape. 

The  face  formed  by  the  cutting  plane  is  called  the  inclined 
section. 

The  other  faces,  that  is,  the  lateral  faces,  are  quadrilaterals, 
either  trapezoids  or  trapeziums. 

Gi\  e  the  name  of  each  face  of  your  pyramid. 
Are  any  of  the  faces  equal  ? 


CHAPTER    XIV 


Testing  a  Curved  Surface 
CURVED    SURFACES    AND    LINES 

I.  WE  will  now  begin  the  subject  of  curved  surfaces  and 
curved  lines.  Curved  means  "crooked,"  or  "bent  without 
corners."  If  you  try  to  hold  the  straight  edge  of  a  ruler 
against  some  surfaces,  you  will  find  that  you  can  do  so  only 
with- certain  positions  of  the  ruler,  sometimes  with  no  position 
whatever;  such  surfaces  are  curved.  Probably  you  can  see 
objects  about  the  room  having  curved  surfaces,  some  of  which 


88  OBSERVATIONAL   GEOMETRY 

you  can  test  with  a  ruler's  edge.  Perhaps  you  can  find  some 
curved  surfaces  upon  which  you  can  place  the  ruler's  edge  in 
certain  directions,  but  not  in  all  directions,  as  you  can  in  the 
case  of  planes;  but  with  most  curved  surfaces  there  will  be 
no  direction  whatever  in  which  you  can  hold  the  straight 
edge  upon  them. 

In  the  case  of  water,  if  the  amount  is  not  great,  the  surface  is  considered  to 
be  a  plane;  but  large  bodies  of  water  even  when  at  rest  must  have  curved  sur- 
faces ;  for  these  have  the  form  of  the  earth's  surface,  which  is  a  curve. 

Curved  edges,  that  is,  curved  lines,  are  formed  by  curved 
surfaces  meeting  other  surfaces,  curved  or  plane.  Thus  a 
plane  surface  can  have  a  curved  edge. 

2.    Of  all  faces  bounded  by  curves  the  commonest  is  the 
circle. 

Circle  means  "a ring." 


Make  a  dot  on  paper;  then,  with  the  aid  of  a  ruler  draw 
from  the  dot  a  number  of  straight  lines  each  two  centimetres 
(or  I  in.)  long.  If  you  make  these  lines  quite  close  together, 
you  will  see  that  their  ends  can  be  connected  by  a  certain 
curved  line;  that  line  is  called  the  circumference  (cir-cum'- 
fer-ence)  of  the  circle. 

Circumference  means  "  carrying  around." 

Each  of  the  straight  lines  is  called  a  radius  (ra'-di-us)  of 
the  circle;  radius  means  "a  ray." 

The  point  from  which  you  drew  the  equal  straight  lines  is 
called  the  centre  of  the  circle. 


CURVED   SURFACES  AND   LINES 


89 


The  circle  is  the  face  itself,  and  is  defined  as  a  face  bounded 
by  a  curved  line  all  parts  of  which  are  equally  distant  from  a 
point  within  called  the  centre. 

The  word  "  circle  "  is  sometimes  applied  to  the  curve  which  bounds  it,  just 
as  the  word  "  ring  "  may  refer  to  the  curved  boundary  or  to  the  enclosed  space ; 
but  to  be  accurate  you  should  call  the  curved  line  the  circumference  and  the 
face  itself  the  circle. 

The  radii  measure  the  distance  from  the  centre  to  the  cir- 
cumference and  are  therefore  all  equal  to  one  another. 

An  arc  is  any  part  of  a  circumference  ;  arc  means  "  a  bow." 


An  Arc 

Knowing  that  the  centre  of  a  circle  is  equally  distant  from  all  parts  of  the 
circumference,  can  you  decide  from  the  picture  of  the  archer  whether  or  not 
his  right  hand  is  at  the  centre  of  the  circle  of  which  his  bow  forms  an  arc  ? 


An  Archer 


90  OBSER  VA  TIONA  L   GEOME  TR  Y 

A  diameter  (di-a'-me-ter)  is  a  straight  line  drawn  through 
the  centre  of  a  circle  and  bounded  by  the  circumference. 
Every  diameter  divides  the  circle  and  the  circumference  into 
two  equal  parts. 


A  Diameter 

How  many  radii  does  it  take  to  make  one  diameter  ? 
Are  all  the  diameters  of  a  circle  equal  to  one  another  ? 

The  circle  and  the  rectangle  are  the  commonest  shapes  in  manufactured 
things.  Probably  you  can  see  many  objects  about  you  which  are  made  in 
these  shapes  ;  but  in  nature  the  circular  is  by  far  the  commonest  shape 
of  all. 


Field  Artillery 


CURVED  SURFACES  AND  LINES  91 

In  the  picture  of  the  cannon  wheel,  to  what  part  of  a  circle  does   the  tire 
correspond  ? 
The  spokes? 
The  axle  ? 

Can  you  calculate  the  angle  which  each  spoke  makes  with  the  one  next  to  it? 
Are  the  spokes  so  arranged  that  each  forms  with  another  a  diameter  ? 

3.  Railroad  Curves.  Curved  lines  can  be  parallel  to  one 
another;  and  then,  as  in  the  case  of  parallel  straight  lines, 
their  distance  apart  remains  unchanged. 

The  rails  of  a  track  are  perhaps  the  most  familiar  instance 
of  parallel  curved  lines. 

Can  you  think  of  other  instances? 

In  one  respect  a  curved  line  differs  completely  from  a 
straight  line.  You  have  seen  that  a  straight  line  retains  one 
direction  throughout  its  extent.  Now  a  curved  line  changes 
its  direction  throughout  its  extent. 


Thus  in  the  annexed  curve,  if  you  place  your  pencil  point  at  A  and 
follow  the  curve  around  through  B,  C,  and  D,  back  to  A,  you  will  find 
that  your  pencil  is  always  changing  its  direction.  At  C  it  is  moving  in 
the  opposite  direction  to  that  in  which  it  started  ;  at  D  in  the  opposite 
direction  to  that  in  which  it  was  moving  at  B ;  and  finally  it  regains  its 
first  direction  when  it  gets  back  to  its  starting-point  at  A. 

Notice,  also,  that  if  you  compare  the  direction  of  the  curve 
with  the  direction  of  some  straight  line,  such  as  XY,  there 
must  have  been  a  point  where  the  curve  and  the  straight 


92 


OBSERVATIONAL    GEOMETRY 


line  had  the  same  direction;  but  while  the  straight  line  con- 
tinued in  that  direction,  the  curved  line  instantly  changed  to 
a  new  direction. 


An  important  use  of  th-s  truth  is  made  in  laying  railroads  so  that  trains 
may  change  their  direction  without  running  off  the  track.  Suppose  that 
the  track  AB  is  curved,  and  at  B  the  direction  is  to  be  changed  to  a 
straight  line.  The  men  who  are  planning  the  route  find  the  direction  of 
the  curve  at  B  by  drawing  a  radius  to  that  point  and  making  a  perpen- 
dicular BC,  along  which  they  lay  the  track.  So  when  a  train  from  A 
reaches  B,  it  keeps  the  direction  it  then  has  and  goes  in  a  straight  line 
towards  C.  If  at  any  point,  as  D,  another  curve  has  the  same  direction 
as  BC,  and  both  tracks  are  laid,  towards  E  as  well  as  towards  C,  a  train 
from  A  on  reaching  D  can  go  in  either  direction  ;  but  a  "  switch  "  pre- 
vents this  by  cutting  off  the  track  which  is  not  to  be  used. 

On  the  right  of  the  picture  you  can  see  the  switch-house  from  which 
the  tracks  are  controlled. 


CURVED   SURFACES   AND   LINES 


93 


Railroad  Junction 


4.  Three  Ways  of  drawing  a  Circumference.  Later  on  we 
shall  have  much  more  to  say  about  circles;  but  now  you 
need  learn  only  enough  to  help  you  in  drawing  figures 
based  upon  circles. 

First,  we  will  see  how  circular  faces   can  be  marked  off. 

There  are  many  ways  of  doing  this,  three  of  which  you 
may  use. 

First,  a  circumference  can  be  drawn  with  an  instrument 
called  "a  pair  of  compasses"  or  simply  "compasses."  A 
pair  of  compasses  is  a  pair  of  dividers  one  of  whose  prongs 
carries  at  the  end  a  pencil  or  a  pen. 


94 


OBSER  VA  TIONAL   GEOME  TR  Y 


To  use  the  compasses,  place  the  pointed  end  firmly  on  the  paper,  and 
revolve  the  pencil  end  lightly  until  the  line  it  makes  comes  around  to  its 
starting-point.  The  place  where  the  fixed  end  rests  is  the  centre  of  the 
circle;  the  distance  between  the  points  of  the  instrument  is  the  length  of 
the  radius ;  and  as  that  distance  does  not  change  during  the  revolution, 
the  line  you  have  made  is  the  circumference  of  the  circle,  and  the  space 
enclosed  is  the  circle  itself. 

With  practice  you  will  find  that  you  can  use  the  compasses  best  if  you 
hold  them  between  the  thumb  and  forefinger,  and  press  only  firmly 
enough  to  keep  the  fixed  end  from  sliding  from  its  place  at  the  centre. 


Drawing  a  Circle  with  a  Pair  of  Compasses 

Secondly,  if  you  have  no  compasses,  you  can  draw  a  cir- 
cumference with  the  aid  of  a  string  which  has  a  loop  at  each 
end.  The  length  of  the  string  will  be  the  radius. 

Place  a  pin  in  one  loop  where  the  centre  of  your  circle  is  to  be ;  then 
insert  the  pencil  point  in  the  other  loop,  draw  the  string  tight  against  the 
paper,  and  move  the  pencil  around.  Its  point  will  rrmrk  the  circumfer- 
ence. 


CURVED  SURFACES  AND  LINES 


95 


Drawing  a  Circle  with  a  String 

This  method  is  a  convenient  one  to  use  in  case  you  wish  to  make  a  very 
great  circle ;  for  example,  on  the  ground,  where  you  would  use  a  stake,  a 
rope,  and  a  pointed  stick  to  mark  the  circumference. 


Drawing  a  Circle  with  a  Cord 


96  OBSERVATIONAL   GEOMETRY 

Thirdly,  you  can  draw  a  circumference  with  the  aid  of  a 
card,  in  which  two  small  holes  are  made. 


Drawing  a  Circle  with  a  Card 

The  card  is  laid  flat  on  the  paper;  through  one  of  the  holes  a  pin  is 
inserted  where  the  centre  is  to  be  ;  then  the  pencil  point  is  inserted 
through  the  other  hole,  and  the  card  is  revolved  with  the  pencil  point 
marking  the  circumference  as  it  moves. 

This  method  has  one  convenience,  that  since  the  distance  between  the 
holes  in  the  card  is  the  length  of  the  radius,  a  series  of  holes  may  be  made 
at  various  distances  noted  on  the  card,  thus  avoiding  constant  measuring 
of  radius  lengths. 


CHAPTER    XV 


The  Middle  Tower 
A    CYLINDER 

I.    IN  the  picture  of  "The  Tower  of  London"  you  can 
see  two  examples  of  what  is  called  "  a  round  body." 

This  is  a  cylinder  (cyl'-in-der),  a  word  meaning  "  a  roller." 
A  cylinder  has  three  surfaces,  two  of  them  being  equal  par- 
allel planes  called  the  bases  of  the  cylinder,  and  the  third 
being  a  curved  surface.  The  edges  of  a  cylinder  are  curved, 
and  it  has  no  corners.  The  cylinder  is  a  common  form  in 
manufactured  things ;  probably  you  can  think  of  many  objects 
which  have  this  shape,  such  as  pencils,  parts  of  machines,  etc. 

7 


OBSERVATIONAL   GEOMETRY 
We  will  now  make  a  model  of  a  cylinder. 


2,   The  diagram  will  need  paper  16  cm.  X  15  cm.  (or  6£  in.  X  6  in.). 

First  draw  the  rectangle  A  BCD  with  sides  15  cm.  7  mm.  (or  6/j  in.)  and  5 
cm.  (or  2  in.). 

Then,  with  L  and  R  as  centres  and  a  radius  of  25  mm.  (or  I  in.)  draw  circles 
so  that  they  may  just  touch  the  longer  sides  of  the  rectangle.  The  lapels  on 
these  longer  sides  should  be  wedge-shaped  and  broader  than  usual.  In  cutting 


A    CYLINDER  99 

out  the  figure,  be  careful  not  to  separate  the  two  circles  entirely  from  the 
rectangle. 

Paste  first  the  edges  AC  and  BD.  Then  paste  the  other  lapels  on  the  outside 
of  the  circular  ends  L  and  R  ;  these  edges  should  then  be  strengthened  with  a 
thin  strip  of  paper;  or  you  can  make  the  circles  a  little  smaller  so  that  they 
may  fit  inside  the  figure,  and  then  cover  them  with  circles  of  full  size. 

3.   1.    How  does  this  cylinder  resemble  a  prism  ? 

2.  What  is  the  smallest  number  of  faces  which  can  bound  a  prism  ?     What 

shape  has  its  base  ? 

3.  What  is  a  straight  line  ?     Are  there  various  kinds  of  straight  lines? 

4.  What  is  a  curved  line  ?     Are  there  various  kinds? 

5.  What  is  a  circumference  ? 

6.  What  is  a  circle? 

7.  What  is  an  arc? 

8.  What  is  the  centre  of  a  circle? 

9.  What  is  the  difference  between  a  circle  and  a  circumference  ? 

10.  What  is  the  difference  between  a  diameter  and  a  radius  ? 

11.  What  was  the  shape  of  the  quadrilateral  which  you  bent  so  as  to  form  the 

curved  surface  of  the  cylinder? 

12.  Which  two  edges  of  the  quadrilateral  join  the  curved  edge  of  the  bases? 

13.  How  do  those  edges  compare  in  length  with  the  circumference  of  the  bases  ? 

14.  Which  two  edges  of  the  quadrilateral  are  equal  to  the  distance  between  the 

bases  of  the  cylinder  ? 

15.  This  quadrilateral  has  formed  a  curved  surface:  what  is  a  curved  surface? 

16.  How  can  you  test  whether  a  surface  is  curved  or  not  ? 

17.  Can  a  straight  line  be  drawn  on  the  curved  surface  of  a  cylinder  ? 

18.  Can  several  straight  lines  be  so  drawn  ?     If  so,  how  do  their  directions 

compare  with  each  other  ? 

19.  Can  you  imagine  your  cylinder  to  be  fitted  exactly  into  a  cubical  box  ?   If  so, 

what  would  be  the  dimensions  of  the  interior  of  the  box  ? 

20.  Can  you  hold  a  ruler  with  its  edge  against  the  curved  surface  of  your  cyl- 

inder in  such  a  position  as  will  show  that  a  straight  line  could  not  be 
drawn  in.  that  direction  on  the  surface  ? 

4.  The  length  of  the  circumference  of  any  circle  is  about 
three  times  the  length  of  its  own  diameter:  it  is  really  a  little 
more  than  three  times ;  three  and  one-seventh  times  would  be 
more  exact. 

You  can  test  this  in  two  ways.  First  you  refer  back  to  the  diagram 
from  which  you  made  the  cylinder:  — 

21.  What  is  the  length  of  the  diameter  of  one  of  the  circles  ? 

22.  What  is  the  length  of  the  edge  of  the  rectangle  which  is  to  be  bent  around 

so  as  to  meet  the  circumference  of  the  circle? 


ioo  OBSERVATIONAL   GEOMETRY 

23.  How  many  times  do  you  find  one  length  to  be  contained  in  the  other? 

Secondly,  you  can  make  measurements  on  the  surface  of  the  completed 
cylinder  by  bending  a  tape  measure,  or  a  narrow  strip  of  paper,  around 
the  curved  surface  close  to  the  base. 

24.  About  what  is  the  length  of  a  circumference  whose  diameter  is  7  cm.  long  ? 

5.  The  area  of  a  circle  is  about  three-fourths  of  the  area  of 
a  square  which  has  an  edge  equal  to  a  diameter  of  the  circle. 
Thus  in  the  annexed  figure  the  circle  occupies  about  three- 
fourths  of  the  square;  and  the  parts  which  lie  outside  the 
circle,  at  the  corners  of  the  square,  together  make  about  one- 
fourth  of  the  square. 


We  will  test  this  by  an  experiment,  and  at  the  same  time  find  the 
volume  of  the  cylinder. 

Make  another  cylinder,  leaving  out  one  of  the  bases,  and  take  the  cube 
you  have  used  for  measuring. 

First  place  the  bases  of  the  two  figures  together,  and  see  that  the 
diameter  of  the  cylinder  is  equal  to  the  edge  of  the  square. 

Then  place  the  two  figures  on  a  horizontal  plane,  and  with  the  aid  of  a 
ruhr  resting  across  their  tops  see  that  their  heights  are  equal, 

Then  experiment  by  filling  them  with  sand  or  water,  etc.  You  will 
find  that  it  takes  four  fillings  of  the  cylinder  to  make  three  of  the  cube; 
or  if  you  fill  the  cylinder  once,  and  pour  the  contents  into  the  cube,  the 
level  inside  the  cube  will  be  three-fourths  of  the  whole  height. 

Now,  since  the  two  figures  have  the  same  height,  the  difference  in  their 
volumes  depends  on  the  difference  in  the  areas  of  their  bases.  So  the 
circular  base  of  the  cylinder  is  three-fourths  of  the  square  base  of  the 
cube. 


A    CYLINDER  Kscfc 

25.  What  is  the  length  of  an  edge  of  your  cube  ? 

26.  What  is  the  length  of  the  diameter  of  the  base  of  your  cylinder? 

27.  What  is  the  area  of  the  base  of  your  cube  ? 

28.  What  is  the  area  of  the  base  of  your  cylinder  ? 

29.  What  is  the  volume  of  your  cube  ? 

30.  What  is  the  volume  of  your  cylinder  ? 

31.  What  is  the  area  of  the  rectangle  which  was  bent  to  form  the  curved  surface 

of  your  cylinder  ? 

32.  What,  then,  is  the  area  of  the  curved  (or  lateral)  surface  of  your  cylinder  ? 

33.  How  would  you  find  the  lateral  surface  of  a  cylinder  given  to  you  ready 

made  ? 

34.  If  you  knew  the  area  of  the  base  of  a  cylinder  and  the  height,  how  would 

you  find  the  volume  ? 

85.    What  is  the  volume  of  a  cylinder  whose  height  is  8  cm.,  and  the  area  of 
whose  base  is  20  sq.  cm.  ? 

36.  What  is  the  volume  of  the  greatest  cylinder  which  can  be  placed  in  a 

cubical  box  6  inches  deep  ? 

37.  How  many  square  inches  are  there  in  the  entire  surface  of  a  cylinder  whose 

lateral  surface  is  formed  from  a  rectangle  5  in.  long  and  4  inches  wide, 
and  whose  bases  are  circles  ? 

38.  What  is  the  volume  of  the  above  cylinder  ? 

The  length  of  a  circumference  is  about  three  (more  nearly  3^)  times  the 
length  of  its  diameter. 

The  area  of  a  circle  is  about  three-fourths  of  the  area  of  the  square  on 
its  diameter. 

The  area  of  the  lateral  surface  of  a  cylinder  is  the  product  of  the  cir- 
cumference of  its  base  and  the  distance  on  the  surface  bet-ween  the  bases. 

The  volume  of  a  cylinder  is  the  product  of  the  area  of  its  base  and 
altitude. 


CHAPTER  XVI 


Mount  Fuji 


A    CONE 

i.  IN  the  picture  of  Mount  Fuji  you  have  another  ex- 
ample of  a  round  body.  This  is  a  cone,  a  word  which  origi- 
nally meant  "  a  peak,"  that  is,  the  top  of  a  mountain.  A 
cone  has  two  surfaces,  one  plane  and  the  other  curved.  The 
plane  surface  is  the  base  of  the  cone,  and  is  bounded  by  a 
curved  line.  The  curved  surface  begins  at  a  point  called  the 
apex  or  vertex  of  the  cone,  and  extends  to  the  base. 
We  will  now  make  a  model  of  a  cone. 


A    CONE 


2.  The  diagram  will  need  paper  12  cm.  X  n  cm.  (or  5  in.  X  4^  in.). 

First  draw  the  angle  ACB,  160°. 

Then  with  the  vertex  C  as  a  centre,  and  a  radius  of  5  cm.  6  mm.  (or  2\  in.) 
draw  the  arc  AB  between  the  sides  of  the  angle. 

Then  with  L  as  a  centre  and  a  radius  of  25  mm.  (or  i  in.)  draw  a  circle  just 
touching  the  arc. 

Make  wedge-shaped  lapels  on  the  arc,  and  be  careful  not  to  cut  the  circles 
entirely  apart.  The  lapels  should  be  pasted  on  the  outside  of  the  circular  base, 
and  the  edge  should  be  strengthened  with  a  thin  strip  of  paper  or  a  second 
circle,  as  you  did  in  the  case  of  the  cylinder. 

3.  The  plane  figure  which  you  bent  so  as  to  form  the 
curved  surface  of  the  cone  is  called  a  sector ;  it  is  a  part  of  a 
circle  bounded  by  two  radii  and  an  arc. 


104 


OB  SEX  VA  TIONA  L   GEOME  TR  Y 


The  height  of  a  cone  is  the  perpendicular  distance  from 
the  apex  to  the  base,  as  AP  (see  figure).  In  the  cone  you 
have  made,  this  line  passes  through  the  centre  of  the  base. 


Testing  the  Surface  of  a  Cone 


The  slant  height  of  a  cone  is  the  distance  from  the  apex  to 
the  circumference  of  the  base,  as  AB,  AC,  AD,  etc.  (see 
figure)  ;  it  is  measured  in  a  straight  line,  and  is  the  only 
possible  case  of  straight  lines  on  the  curved  surface  of  the 
cone,  as  you  can  see  by  holding  a  ruler's  edge  against  the 
surface.  In  your  cone  the  slant  heights  are  all  equal. 

In  the  picture  of"  Cloud  Cap  Mountain"  we  have  an  ex- 
ample of  a  part  of  a  cone  called  a  frustum.  The  frustum  of  a 


A    CONE  105 


cone  is  the  part  which  lies  between  the  base  and  a  plane 
which  cuts  the  cone  parallel  to  the  base. 

The  part  cut  off  above  the  plane  is  a  smaller  cone. 


Cloud  Cap  Mountain 

4.  The  area  of  the  curved  (lateral)  surface  of  your  cone  is 
equal  to  the  length  of  the  circumference  of  the  base  multi- 
plied by  one-half  the  slant  height. 

First,  we  will  find  the  length  of  the  circumference  by  calculation  and 
test  the  answer  by  measurement:  — 

1.  What  is  the  length  of  the  diameter  of  the  base  ? 

2.  By  what  will  you  multiply  the  diameter  in  order  to  find  the  length  of  the 

circumference  ? 

3.  What,  then,  is  the  length  of  the  circumference? 


io6 


OBSERVATIONAL   GEOMETRY 


Now  measure  the  length  of  the  circumference  with  a  tape  or  narrow 
strip  of  paper,  and  see  if  the  two  results  agree. 

Next,  we  will  find  the  slant  height  from  the  diagram  from  which  you 
made  the  cone,  and  test  the  answer  by  measurement :  — 

4.  What  line  on  the  diagram  corresponds  to  the  slant  height  ? 

5.  What  is  its  length? 

Now  measure  the  slant  height  on  the  surface  of  the  cone,  beginning  at 
the  apex.  Remember  that  we  wish  to  measure  a  straight  line,  although 
on  a  curved  surface ;  and  the  only  possible  straight  lines  on  the  lateral 
surface  of  a  cone  are  those  which  pass  through  the  apex  or  would  do  so 
if  prolonged. 

Lastly,  we  will  find  the  area  of  the  lateral  surface  by  multiplying  the 
length  of  the  circumference  by  one-half  the  slant  height.  The  answer  is 
about  44  sq.  centimetres,  or,  if  you  have  used  English  measurements, 
about  7  sq.  inches. 


A  Medicine-Man's  Lodge 


5.  The  volume  of  a  cone  is  equal  to  one-third  of  the 
volume  of  a  cylinder  whose  base  and  height  are  equal  to  those 
of  the  cone. 


A    CONE  107 

We  will  test  this  by  an  experiment.  Make  another  cone,  leaving  out 
the  base,  and  take  the  cylinder  you  have  used  for  measuring. 

First,  place  the  bases  of  the  two  figures  together  and  see  that  they  are 
equal.  Then  with  the  aid  of  a  ruler  resting  across  their  tops  see  that 
their  heights  are  equal.  Then  compare  the  volumes  of  the  figures  by 
filling  them  with  sand  or  water,  etc.  You  will  find  that  it  takes  three 
fillings  of  the  cone  to  make  one  of  the  cylinder. 

6.  Now  the  volume  of  a  cylinder  is  equal  to  the  area  of  its  base  multiplied  by 

its  height :  what  then  is  the  volume  of  your  cone  ? 

7.  The  picture  of  "  the  medicine-man's  lodge  "  represents  a  cone-shaped  tent 

with  a  diameter  and  height  each  about  15  feet.     The  length  of  the  poles 

from  the  top  to  the  outer  edge  is  about  17  feet. 
How  many  square  feet  of  material  was  needed  to  make  this  lodge? 
How  many  cubic  feet  does  it  contain  ? 

8.  What  is  the  volume  of  a  cone  whose  height  is  6  cm.,  and  the  area  of  whose 

base  is  20  sq.  cm.  ? 

9.  What  is  the  volume  of  a  cone  whose  height  is  12  inches,  and  the  diameter 

of  whose  base  is  8  inches  ? 

10.   If  a  cone  and  a  cylinder  have  equal  bases,  but  the  cone  is  three  times  as 
tall  as  the  cylinder,  how  do  their  volumes  compare  ? 

The  area  of  the  lateral  surface  of  a  cone  is  one-half  the  product  of  its 
circumference  and  slant  height. 

circumference  of  base  x  slant  height 
Lateral  surface  cone  =  —  — - — 

circumference  of  base 
= ^—    -  x  slant  height 

slant  height 
=  circumference  of  base  x  - 

The  volume  of  a  cone  is  one-third  the  product  of  the  area  of  its  base  and 
altitude. 

base  x  altitude      base  altitude 

Volume  cone  = =  —    x  altitude  =  base  x 

33  3 


CHAPTER   XVII 


TO  VALVE 

Watt's  Governor 


SOLIDS    OP    REVOLUTION. — THE    SPHERE 

I.  JUST  as  a  flame  at  the  end  of  a  stick  which  is  whirled 
rapidly  around  looks  like  a  continuous  circle  of  fire,  so 
various  plane  figures  when  revolved  about  an  axis  give  the 
appearance  of  solid  bodies. 

Thus  in  "Watt's  Governor"  the  triangle  formed  by  the  two  rods  of  the 
governor  which  carry  the  balls  looks  like  a  cone  when  the  engine  is  in 
motion,  and  the  hexagon  EFMNLK  looks  like  two  frustums  of  cones 
with  their  larger  bases  together. 

Hence  certain  solid  figures  are  called  "  solids  of  revolu- 
tion ; "  for  we  can  imagine  them  as  formed  or  generated  by 
the  revolution  of  plane  figures.  There  are  three  principal 


SOLIDS    OF  REVOLUTION.— THE  SPHERE 


109 


solids   of  revolution,   two   of  which- — the   cylinder  and    the 
cone  —  you  have  already  studied. 

A  cylinder  is  generated  by  the  revolution  of  a  rectangle 
about  one  of  its  edges. 


Thus  the  rectangle  ABCD  revolving  about  CD  as  an  axis 
forms  a  cylinder  whose  height  is  CD,  and  whose  bases  are 
circles  with  radii  equal  to  BD. 

A  cone  is  generated  by  the  revolution  of  a  right  triangle 
about  one  of  the  sides  of  the  right  angle.  Thus  the  triangle 
ACB,  revolving  about  BC  as  an  axis,  forms  a  cone  whose 
height  is  BC,  slant  height  AB,  and  whose  base  is  a  circle  with 
a  radius  equal  to  CA. 


no  OBSERVATIONAL   GEOMETRY 

We  will  now  consider  the  third  solid  of  revolution.  If  you 
spin  a  coin  on  its  edge,  you  will  see  that  it  takes  the  appear- 
ance of  a  solid  figure  different  from  those  which  we  have 
hitherto  studied.  The  coin  is  a  circle  which  has  revolved 
about  one  of  its  diameters;  but  the  same  effect  would  be 
produced  if  only  a  semi-circumference  were  to  revolve  on  its 
diameter. 


2.   This  figure  is  called  a  sphere ;  it  is  a  very  common 
shape  in  nature,  manufactures,  and  the  arts. 

The  word  is  derived  from  a  Greek  word  meaning  "  a  ball,  or  globe." 

The  surface  of  a  sphere  is  curved,  and  all  parts  of  it  are 
equally  distant  from  a  point  within  the  sphere,  which  is  the 
centre. 

A  radius  of  a  sphere  is  a  straight  line  drawn  from  the 
centre  to  the  surface. 

A  diameter  of  a  sphere  is  a  straight  line  drawn  through  the 
centre  and  bounded  at  each  end  by  the  surface.  Thus  a 
diameter  is  equal  to  two  radii. 

All  the  radii  of  a  sphere  are  equal,  and  all  the  diameters 
are  equal. 

Poles  of  a  sphere  are  the  ends  of  any  diameter,  and  are 
therefore  points. 


SOLIDS  OF  REVOLUTION.  — THE  SPHERE 


in 


The  word,  as  used  in  geometry,  is  derived  from  a  Latin  word  meaning  "  a 
pivot."  Thus  the  poles  of  the  earth  are  the  two  points  at  the  ends  of  the  diam- 
eter on  which  as  an  axis  the  earth  revolves. 


No  straight  line  can  be  drawn  on  the  surface  of  a  sphere, 
as  you  can  easily  see  by  trying  to  hold  a  ruler's  edge  against 
the  surface.  Circumferences  of  circles,  however,  can  be 
drawn  ;  and  the  circles  are  of  two  kinds,  —  great  circles  and 
small  circles. 


SOUTH  POLE 

SPHERE 
Poles  and  Axis 


A  great  circle  has  the  same  radius  and  diameter  as  the 
sphere  itself;  it  is  the  greatest  possible  circle  whose  circum- 
ference can  be  drawn  on  the  surface  of  the  sphere. 


112 


OBSERVATIONAL    GEOMETRY 


The  equator  and  meridians  of  longitude  are  examples  of  great  circles  of  the 
earth,  the  meridians  being  semicircles. 


CCJ^ 


SOUTH  POLE 

Parallels  and  Meridians 

A  small  circle  is  a  circle  whose  radius  is  less  than  the 
radius  of  the  sphere. 

Parallels  of  latitude  are  examples  of  small  circles  of  the  earth. 

Every  great  circle  divides  the  sphere  into  two  equal  parts 
called  hemispheres;  the  word  means  "half  a  sphere."  The 
hemisphere  is  a  common  shape  in  the  domes  of  buildings. 

In  the  picture  of  Jerusalem  you  can  see  two  hemispherical  domes. 
On  the  dome  of  the  Greek  Church  how  many  great  circles  are  indicated? 
How  many  small  circles  ? 

Of  which  kind  is  the  circle  which  bounds  the  base  of  this  dome  ? 
On  the  dome  of  the  Church  of  the  Holy  Sepulchre  of  which  kind  are  the 
circles  indicated  ? 

Zones  are  portions  of  the  surface  bounded  by  the  circum- 
ferences of  parallel  circles. 

*The  word  "  zone  "  is  derived  from  a  Greek  word  meaning  "  a  belt." 

The  circumferences  which  bound  the  zone  are  called  the 
bases  of  the  zone. 

The  Torrid  and  Temperate  Zones  on  the  earth's  surface  are  examples  of 
zones  of  two  bases.  The  bases  of  the  Torrid  Zone  are  the  Tropic  of  Cancer 


SOLIDS  OF  REVOLUTION. -THE  SPHERE          113 

and  the  Tropic  of  Capricorn.  The  Frigid  Zones  are  examples  of  zones  of  one 
base.  The  one  base  of  the  North  Frigid  Zone  is  the  Arctic  Circle ;  but  we  can 
Imagine  the  other  base  to  lie  outside  the  earth,  at  the  North  Pole. 


Church  of 
the  Holy 
Sepulchre 


Greek 

Church 


Churches  in  Jerusalem 

3.   The  area  of  the  surface  of  a  sphere  is  exactly  equal  to 
the  area  of  four  great  circles. 


H4  OBSERVATIONAL   GEOMETRY 

Thus  if  the  diameter  is  5  cm.,  the  area  of  a  great  circle  is  about  19^  sq.  cm. 
and  the  area  of  the  sphere  is  about  78  sq.  cm. 


ZONES 


The  surface  of  a  sphere  is  also  exactly  equal  to  the  lateral 
(curved)  surface  of  a  cylinder  which  will  just  contain  the 
sphere. 

An  important  use  is  made  of  this  truth,  in  drawing  maps  so  as  to  represent 
the  curved  surface  of  the  earth  by  a  flat  map  on  which  the  parallels  arid  meridians 
appear  as  straight  lines. 

The  map  is  drawn  on  the  lateral  surface  of  a  cylinder  which  is  then  unrolled 
so  as  to  form  a  rectangle,  thus  reversing  the  process  by  which  you  made  your 
cylinder. 

This  is  called  "  drawing  maps  by  Mercator's  projection." 


4.   Volume  of  a  Sphere.     If  we  consider  a  circumference 
to  be   three  times  as  long  as  its  diameter,  the  volume  of  a 


SOLIDS   OF  REVOLUTION.— THE  SPHERE          115 

sphere  can  be  calculated  by  multiplying  the  diameter  by 
itself,  multiplying  again  by  the  diameter,  and  then  dividing 
by  2. 

Thus,  if  the  diameter  of  a  sphere  is  5  cm.  long,  the  volume  of  the  sphere  is 
5  X  5  X  5  -r-  2,  or  62^  cubic  centimetres. 

Find  the  areas  of  the  surfaces  of  the  following  spheres  :  — 

1.  Diameter  4  cm. 

2.  "         6  " 

3.  "         8  inches. 

4.  Radius       4  cm. 

5.  "  6  inches. 

Find  the  volumes  of  the  following  spheres :  — 

6.  Diameter  2  cm. 

7.  "         3  " 

8.  "         4  inches. 

9.  Radius  I  cm. 
10.        "      2  inches. 

The  area  of  the  surface  of  a  sphere  is  four  times  the  area  of  a  great 
circle. 

The  wlum*  of  a  sphere  =  diameter  x  diameter  x  ***"*" 


CHAPTER   XVII  I 

FIGURES    FOR    PRACTICE 

THE  general  name  for  a  figure  of  three  dimensions  is 
polyedron,  which  means  "  having  many  faces." 

The  following  figures  are  more  difficult  to  construct  and 
observe.  Many  of  them  are  combinations  or  parts  of  figures 
which  you  have  already  made.  Some  resemble  crystal  forms 
which  occur  in  nature. 

Three  are  regular  polyedrons  ;  that  is,  their  faces  are  equal 
regular  polygons,  and  their  diedral  angles  are  equal. 

Only  five  regular  polyedrons  are  possible,  of  which  you  have  already  constructed 
two,  the  cube  and  the  equilateral  triangular  pyramid. 

When  you  have  completed  a  figure,  you  should  examine  it 
carefully,  and  see  if  you  can  answer  the  following  questions  : 

1.  Is  the  figure  a  combination  of  smaller  ones  ?     If  so,  what  are  their  shapes  ? 

2.  Is  the  figure  a  part  of  another  ?    If  so,  what  is  the  shape  of  the  other  ?   How 

was  the  division  made  ? 

3.  How  many  faces  has  the  figure? 

4.  Describe  the  shapes  of  the  faces ;  and  if  there  are  several  kinds,  find  the 

number  of  each  kind. 

5.  How  many  edges  has  the  figure  ? 

6.  What  are  the  lengths  of  the  edges  ? 

7.  How  many  corners  or  solid  angles  has  the  figure  ? 

8.  How  many  faces  form  the  solid  angles  ? 

9.  How  many  diedral  angles  has  the  figure  ? 

10.  What  are  the  sizes  of  the  diedral  angles  ? 

11.  How  many  line  angles  are  there  on  the  surface  of  the  figure  ? 

12.  What  are  the  sizes  of  the  line  angles  ? 

13.  What  is  the  volume  of  the  figure  ? 

The  volume  should  be  found  by  experiment.  Before  clos- 
ing the  last  face  fill  the  figure  with  sand,  and  pour  the  contents 
into  a  cube,  where  measurements  can  easily  be  made. 


FIGURES  FOR  PRACTICE 


117 


A   TRUNCATED    TRIANGULAR    PRISM 

The  diagram  will  need  paper  16  cm.  X  15  cm.  (or  6|  in.  X  6  in.). 

The  construction  can  be  seen  from  the  special  figure. 

A,  B,  and  C  are  equal  squares  with  edges  5  cm.  (or  2  in.)  long. 

From  the  upper  corners  of  B  lines  are  drawn  to  the  middle  points  of  the  outer 
edges  of  A  and  C. 

With  two  edges  equal  to  these  lines  an  isosceles  triangle  D  is  constructed  on 
the  upper  edge  of  B. 

L  is  an  equilateral  triangle  constructed  on  the  lower  edge  of  B. 


OBSERVATIONAL   GEOMETRY 


A  QUADRANGULAR   PRISM 

The  diagram  will  need  paper  20  cm.  5  mm.  X  15  cm.  (or  8|-  in.  X  6  in.). 

The  surface  consists  of  four  equal  squares  with  edges  5  cm.  (or  2  in.)  long, 
and  two  equal  rhombuses  with  angles  60°  and  120°  and  edges  5  cm.  (or  2  in.) 
long. 


FIGURES  FOR  PRACTICE 


119 


A  RHOMBIC   PRISM 

The  diagram  will  need  paper  20  cm.  X  14  cm.  (or  8  in.  X  6  in.). 
The  surface  consists  of  equal  rhombuses  with  angles  60°  and  120°  and  edges 
5  cm.  (or  2  in.)  long. 

These  prisms  are  to  be  used  for  comparison  with  the  cube. 
How  do  the  three  figures  compare:  — 

1.  In  the  number  of  edges  ? 

2.  In  the  total  length  of  edges  ? 

3.  In  the  number  of  faces  ? 

4.  In  the  total  area  of  their  surfaces  ? 

5.  In  their  volumes  ? 


120 


OBSERVATIONAL   GEOMETRY 


THE    REGULAR   OCTAEDRON 

The  diagram  will  need  paper  18  cm.  X  14  cm.  (or  7^  in.  X  6  in.). 

ABC 'and  DEFarz  equilateral  triangles  with  edges  I  dcm.  (or  4  in.)  long; 
D  is  the  middle  point  of  AC.  These  two  triangles  are  each  divided  into  four 
equilateral  triangles  by  joining  the  middle  points  of  the  edges. 

This  figure,  the  third  in  the  series  of  regular  polyedrons,  is  called  an  octae- 
dron,  because  it  has  eight  faces. 


FIGURES  FOR  PRACTICE 


121 


A  B 

THE    REGULAR   ICOSAEDRON 


The  diagram  will  need  paper  17  cm.  x  8  cm.  (or  7!  in.  X  3  in.)." 

The  construction  can  be  seen  from  the  special  figure. 

A  BCD  is  a  parallelogram  with  angles  60°  and  120°,  and  edges  12  cm.  5  mm. 
and  7  cm.  5  mm.  (or  5  in.  and  3  in.)  long. 

Each  edge  is  divided  into  equal  parts  2  cm.  5  mm.  (or  I  in.)  long.  Then  the 
points  of  division  are  joined  by  three  series  of  parallel  lines  as  in  the  figure, 
thus  dividing  the  parallelogram  into  thirty  equilateral  triangles,  of  which  ten  — 
having  a  side  on  the  top  or  bottom  of  the  parallelogram  —  are  afterwards  re- 
moved, leaving  only  enough  to  form  lapels  of  the  triangles  which  remain. 

The  above  figure,  the  fourth  of  the  series  of  regular  polye- 
drons,  is  called  an  icosaedron  (i-cos-a-e'-dron)  because  it  has 
twenty  faces. 


122 


OBSEK  VA  TIONA  L   GEOME  TR  Y 


B  Z 

THE    REGULAR   DODECAEDRON 


The  diagram  will  need  paper  17  cm.  X  9  cm.  (or  7  in.  X  4  in.). 

The  construction  can  be  seen  from  the  special  figure. 

ABODE  is  a  regular  pentagon,  each  angle  being  108°,  and  each  edge  5  cm. 
(or  2  in.)  long. 

The  five  diagonals  AC,  AD,  etc.,  are  drawn,  forming  a  smaller  pentagon 
within  the  first.  Then  all  the  diagonals  of  the  smaller  pentagon  are  drawn  and 
prolonged  to  the  edges  of  the  larger  one,  thus  forming  five  more  pentagons. 

Next,  the  regular  pentagon  VWXYZ  is  constructed,  V  being  a  vertex  of  one 
«if  the  smaller  pentagons,  and  VW 'one  of  the  edges  prolonged  so  that  F/-Fmay 
be  equal  to  BC.  The  diagonals  are  drawn  as  before. 

This  figure,  the  fifth  and  last  of  the  series  of  regular 
polyedrons,  is  called  a  dodecaedron,  because  it  has  twelve 
faces. 


FIGURES  FOR  PRACTICE 


123 


A   PENTAGONAL   PRISM 


The  diagram  will  need  paper  13  cm.  X  12  cm.  5  mm.  (or  5*  in.  X  5  in.). 
The  faces  consist  of  rectangles  and  regular  pentagons. 
The  rectangles  have  edges  5  cm.  and  2  cm.  5  mm.  (or  2  in.  and  I  in.)  long. 
The  pentagons  have  edges  2  cm.  5  mm.  (or  i  in.)  long,  and  angles  of  108  ° 


124 


OBSERVATIONAL   GEOMETRY 


/vvyvx 

VXAAA/ 


AAAAAAA 


CRYSTAL   OF   SPINEL 

The  diagram  will  need  paper  18  cm.  X  16  cm.  (or  7^  in.  X  6\  in.). 

ABC'vs,  an  equilateral  triangle  with  edges  17  cm.  5  mm.  (or  7  in.)  long.  Each 
edge  is  divided  into  seven  equal  parts  2  cm.  5  mm.  (or  I  in.)  long,  and  lines  are 
drawn  parallel  to  the  edges  of  the  triangle,  connecting  the  points  of  division, 
thus  forming  smaller  equilateral  triangles.  The  lines  which  appear  in  the  dia- 
gram lie  along  the  lines  which  are  in  the  special  figure. 

The  faces  consist  of  equilateral  triangles  with  edges  5  cm.  (or  2  in.)  long; 
rhombuses  with  angles  60°  and  120°,  and  edges  2  cm.  5  mm.  (or  I  in.)  long  ;  and 
trapezoids  with  angles  60°  and  120°,  and  edges  5  cm.  and  2  cm.  5  mm.  (or  2  in. 
and  i  in.)  long. 

This  model  resembles  the  crystal  of  spinel. 


FIGURES  FOR   PRACTICE 


CRYSTAL  OF   COPPER 


The  diagram  will  need  paper  12  cm.  X  8  cm.  (or  5  in.  X  3!  in.). 

The  construction  can  be  seen  from  the  special  figure. 

ABCD  is  a  square  with  edges  7  cm.  5  mm.  (or  3  in.)  long. 

The  edges  are  divided  each  into  three  equal  parts  25  mm.  (or  I  in.)  long ; 
and  parallel  lines  are  drawn  connecting  the  corners  and  other  corresponding 
points  of  division. 

H  is  a  regular  hexagon  constructed  on  the  middle  division  of  the  upper 
edge  of  the  square. 

This  model  resembles  one  of  the  crystal  forms  of  copper. 


126 


OBSERVATIONAL   GEOMETRY 


C  Y  D 

A  TWIN   CRYSTAL  OF  CALCITE 


FIGURES  FOR  PRACTICE  127 

The  diagram  will  need  paper  16  cm.  X  8  cm.  (or  6|  in.  X  3!  in.). 

The  construction  can  be  seen  from  the  special  figure. 

ADCD  is  a  rectangle  with  edges  15  cm.  and  7  cm.  5  mm.  (or  6  in.  and  3  in.), 
divided  into  two  squares  by  the  line  XY. 

The  edges  of  the  squares  are  divided  each  into  three  equal  parts  25  mm.  (or 
i  in.)  long  ;  and  parallel  lines  are  drawn  connecting  the  corners  and  other  cor- 
responding points  of  division. 

This  model  resembles  a  crystal  form  of  calcite,  called  "  a 
twin-crystal,"  as  it  consists  of  two  interpenetrating  cubes. 


PART    II 

POINTS,  LINES,   ANGLES,  POLYGONS,  AND    CIRCLES 


CONSTRUCTIONS,  MENSURATION,  SIMILAR    FIGURES, 
AND   SURVEYING 


CHAPTER    XIX 

POINTS    AND    LINES 

i.  ARRANGEMENTS  of  points  with  reference  to  each  other 
in  the  same  straight  line. 

1.   In  how  many  different  orders  can  two  points  be  placed  with  reference  to 
each  other  in  the  same  straight  line? 

Let  a  and  b  be  two  points. 


I st.  a  can  be  placed  before  b,  as         •  •    

2nd.  b  can  be  placed  before  rt,  as.         • 

Therefore  there  are  two  different  arrangements. 

2.   In  how  many  different  orders  can  three  points  be  arranged  in  one  straight 
line  ? 

Let  a,  b,  and  c  be  the  three  points. 

You  have  seen  by  the  preceding  problem  that  two  points,  a  and  b,  can 
be  arranged  in  two  different  orders 


Now  taking   the  first  group 


introduced  into  it  in  three  different  ways, 


— ,  notice  that  c  can  be 


Likewise  in  the  second  group, 
in  three  different  ways, 


,  c  can  be  inserted 


Therefore  there  are  six  different  arrangements  possible. 


132  OBSERVATIONAL   GEOMETRY 

3.  In  how  many  different  orders  can  four  points  be  arranged  in  one  straight 

line  ? 

Take  one  of  the  groups  of  three  points,  and  introduce  the  fourth  point  into 
it  in  the  various  positions;  then  do  the  same  with  each  of  the  other  groups  of 
tnree  points.  You  will  find  that  there  are  twenty-four  possible  orders  in  all. 

4.  In  how  many  different  orders  can  five  points  be  arranged  in  one  straight 

line  ? 
Write  out  only  one  set  of  groups,  but  calculate  the  total  number. 

5.  Find  the  number  of  orders  for  six  points. 

By  examining  the  method  which  you  have  used  in  the 
preceding  problems,  you  can  obtain  a  rule  by  which  you  can 
calculate  the  total  groups  which  any  number  of  points  would 
form. 

2  points  give    2  groups   =1x2 

3  "          "        6      "         =1x2x3 

4  "          "     24      "         =1x2x3x4. 

Therefore,  to  calculate  the  total  groups  for  any  number  of 
points,  multiply  together  the  numbers  from  I  up  to  and 
including  the  number  of  points. 

6.  Find  by  calculation  the  total  number  of  arrangements  of  7  points  in  the 

same  straight  line. 

7.  What  series  of  numbers,  if  multiplied  together,  would  give  the  total  number 

of  groups  for  10  points  ? 

2.    Points  determined  by  intersecting  straight  lines. 

1.   In  how  many  points  can  two  straight  lines  cut  each  other  ? 
Two  straight  lines  can  cut  each  other  at  only  one  point. 


2.   In  how  many  points  can  three  straight  lines  cut  each  other  ? 

Two  straight  lines  can  cut  each  other  at  one  point,  and  a  third  straight 
line  can  cut  the  other  two  each  at  one  point;  therefore  three  straight 
lines  can  cut  each  other  at  three  points. 


POINTS  AND  LINES  133 


3.  According  to  the  preceding  problem,  three  is  the  greatest  number  of  points 

in  which  three  straight  lines  can  cut  each  other :  can  you  draw  three 
straight  lines  so  that  they  can  cut  each  other  in  only  two  points? 

4.  Can  you  draw  the  three  lines  so  that  they  will  cut  each  other  in  only  one 

point  ? 

5.  Can  you  draw  them  so  that  they  will  not  cut  each  other  in  any  point? 

6.  What  is  the  greatest  number  of  points  in  which  four  straight  lines  can  cut 

each  other  ?     Draw  a  diagram. 

7.  Five  straight  lines  ?     Draw  a  diagram.     [Ans.  10  points.]  . 

8.  Six  straight  lines  ?          "       "        " 

From  the  preceding  problems  you  can  obtain  a  rule  by 
which  you  can  calculate  the  greatest  number  of  points  of 
intersection  which  any  number  of  straight  lines  can  have. 

2  straight  lines  can  have  I  point  of  intersection  =  l 

3  "  "       "       "      3  points "  "  =  1  +  2 

4  "  "       "       "      6     "       "  "  =1  +  2  +  3 

5  "  "       "       "    10     "       "  "  =1+2  +  3+4 

Therefore  to  calculate  the  greatest  possible  number  of 
points  of  intersection  which  a  certain  number  of  straight  lines 
can  have,  add  together  the  series  of  numbers  from  I  up  to 
but  not  including  the  number  of  lines.* 

9.  Find  by  calculation  the  greatest  possible  number  of  points  of  intersection 

among  seven  straight  lines. 
10.    Calculate  the  same  for  eight  straight  lines. 

*  A  still  shorter  method,  shown  in  algebra,  is  to  multiply  the  number  of 
lines  by  I  less  than  the  number  and  divide  the  product  by  2. 

Thus,  10  lines  will  give  —       —  or  45  points  of  intersection. 


134  OBSERVATIONAL   GEOMETRY 

11.  If  the  greatest  possible  number  of  points  of  intersection  among  five  straight 
lines  is  10,  what  would  the  number  be  supposing  that  two  of  the  lines  were 
parallel  ?  Draw  a  diagram. 

3.  To  divide  a  group  of  points  into  two  groups  of  various 
numbers. 

1.  In  how  many  ways  can  two  points  be  divided  into  two  groups?    Ans.  One 

way :  i  —  i . 

2.  Three  points  ?     Ans.  One  way :  i  —  2. 

3.  Four  points  ?     Ans.  Two  ways :  i  —  3,  2  — 2. 

4.  Five  points?     Ans.  Two  ways:  I — 4,  2  —  3. 

5.  Six  points? 

6.  Seven  points  ? 

7.  Eight  points  ? 

8.  Nine  points  ? 

From  the  above  results  a  rule  can  be  formed :  To  find  the 
number  of  ways  in  which  a  group  of  points  can  be  divided 
into  two  groups,  divide  by  2  the  number  of  points  if  it  be  an 
even  number,  or  one  less  than  the  number  of  points  if  it  be  an 
odd  number. 

9.  Calculate  the  number  of  ways  in  which  30  points  can  be  divided  into  two 

groups. 

10.  Calculate  the  same  for  35  points. 

11.  For  48  points. 

12.  For  27  points. 

In  these  problems  you  will  notice  that  you  do  not  raise  the  question  which 
of  the  two  groups  contains  any  particular  point.  Thus  with  three  points 
a,  b,  and  c,  the  groups  a  —  be,  b  —  ac,  and  c  —  ab>  all  come  under  the  head 
of  one  division. 

4.  To  draw  the  greatest  possible  number  of  straight  lines 
between  points. 

1.   Between  two  points  how  many  straight  lines  can  be  drawn  ? 
Let  a  and  b  be  the  points. 


Between  a  and  b  one  straight  line  can  be  drawn,  and  only  one.     The 
straight  line  from  a  to  b  is  the  same  as  that  from  b  to  a. 


POINTS  AND  LINES  135 

2.    Among  three  points  how  many  straight  lines  can  be  drawn  ? 
Let  a,  b,  and  c  be  the  points. 


Between  a  and  b  one  straight  line  can  be  drawn ;  then  the  third  point 
c  can  be  connected  with  each  of  the  other  two  points ;  therefore  three 
straight  lines  can  be  drawn  among  three  points. 

3.  According  to  the  preceding  problem,  three  is  the  greatest  number  of  straight 

lines  which  can  be  drawn  among  three  points :  can  you  place  three  points 
so  that  not  so  many  as  three  straight  lines  can  be  drawn  among  them  ? 

4.  What  is  the  greatest  number  of  straight  lines  which  can  be  drawn  among 

four  points  ?     Draw  a  diagram  for  three  points  and  then  proceed  as  in  the 
second  question. 

5.  Find  the  greatest  number  of  straight  lines  for  five  points,  drawing  a  diagram. 

6.  Do  the  same,  with  six  points. 

You  will  notice  that  in  a  group  of  points,  for  example,  six, 
a  straight  line  can  be  drawn  from  each  of  the  six  to  each  of 
the  other  five,  thus  making  thirty  lines ;  but  the  thirty  must 
be  divided  by  2  to  avoid  counting  each  line  twice,  so  that 
fifteen  is  the  greatest  possible  number  of  different  lines  among 
six  points. 

This  can  be  put  in  the  form  of  a  rule :  To  find  the  great- 
est possible  number  of  different  straight  lines  which  can  be 
drawn  among  a  number  of  points,  multiply  the  number  of 
points  by  I  less  than  that  number,  and  divide  by  2. 

7.  Find  by  calculation  the  greatest  number  of  different  straight  lines  which 

can  be  drawn  among  eight  points. 

8.  Find  the  same  for  eleven  points. 

9.  If  three  of  the  points  in  a  group  lie  in  one  straight  line,  what  change  will 

there  be  in  the  total  number  of  lines  ? 

10.    Can  you  arrange  five  points  so  that  only  one  straight  line  can  be  drawn 
through  them? 


136  OBSERVATIONAL   GEOMETRY 

11.  Can  you  arrange  five   points  so  that  only  five  straight  lines  can  be  drawn 

among  them  ? 

12.  Can  you  draw  a  diagram  showing  how  you  could  plant  seven  trees  so  as 

to  form  six  rows  with  three  trees  in  each  row? 

13.  Can  you  draw  a  diagram  showing  how  you  could  plant  nineteen  trees  so 

as  to  form  nine  rows  with  five  trees  in  each  row?  [Hint:  draw  two  tri- 
angles so  as  to  form  a  six-pointed  star.] 

14.  Can  you  show  how  to  plant  nine  trees  in  ten  rows  with  three  trees  in  each 

row?  [Hint;  begin  by  drawing  a  rectangle  whose  length  is  twice  its 
width;  then  prolong  the  two  shorter  sides  in  opposite  directions,  each  to 
a  distance  equal  to  its  own  length.] 


CHAPTER   XX 

POINTS    OF    INTERSECTION 

I.  To  find  the  number  of  points  of  intersection  of 
straight  lines  which  are  divided  in  various  ways  into  two 
groups,  the  lines  of  each  group  being  parallel  to  one  another. 


LL 


1.   Suppose  the  number  of  lines  to  be  four. 

Four  lines  can  be  divided  into  two  groups  in  two  ways  (see  p.  134),  I  line 
and  3  lines,  or  2  lines  and  2  lines.  If  the  lines  of  each  group  are  parallel,  how 
many  points  of  intersection  will  there  be? 


138  OBSERVATIONAL   GEOMETRY 

You  will  notice  that  each  line  of  a  group  cuts  each  line  of  the  other 
group  in  one  point;  but  the  lines  of  the  same  group  cannot  cut  one 
another :  why  ? 

2.  How  many  points  of  intersection  will  six  straight  lines  make  when  divided 
into  groups  as  in  the  preceding  problem  ? 

—L-         li         III 


II 


7 -U- 


3.  Five  straight  lines  ? 

4.  Eight  straight  lines  ? 

5.  Nine  straight  lines  ? 

6.  If  the  number  of  lines  be  12,  and  the  number  of  points  of  intersection  be 

27,  how  many  lines  are  there  in  each  group  ? 

7.  Can  ii  lines  and  17  lines  each  be  divided  into  two  groups  of  paralleMines 

so  as  to  give  30  points  of  intersection? 

8.  What  numbers  of  lines  can  be  divided  into  groups,  each  of  parallel  lines,  so 

as  to  give  30  points  of  intersection  ? 

9.  Fifteen  lines,  when  divided  in  various  ways  into  two  groups  of  parallel  lines, 

give  the  following  numbers  of  points  of  intersection,—  14,  26,  36,  44,  50, 
54,  56 :  what  do  you  notice  about  the  successive  differences  between  these 
numbers  ? 

10.   The  following  is  a  table  of  the  number  of  points  of  intersection  of  Mnes 
when  divided  in  various  ways  into  two  groups  of  parallel  lines :  — 

3  lines  make  2  points  of  intersection. 

4  «  «  3,  or  4, 

5  "  "  4,  or  6, 

6  «  "  5,        8,  or  9, 

7  "  "  6,       10,  or  12, 

8  "  "  7,       12,       15,  or  16, 

9  "  "  8,       14,       18,  or  20, 

10    "         "     9,       16,       21,       24,  or  25. 

What  do  you  notice  about  the  increase  in  these  numbers  of  points  when  you 
read  the  columns  downwards  ? 


POINTS  OF  INTERSECTION  139 

11.  Continue  the  table  for  n  and  12  lines,  using  the  above  scheme  as  a  guide. 

12.  How  many  lines  parallel  to  a  given  straight  line  can  be  drawn  through  one 

point  ? 

13.  What  is  the  greatest  number  of  lines  which  can  be  drawn  through  four 

points,  parallel  to  a  given  straight  line  ? 

14.  Can  you  place  four  points  so  that  only  one  line  can  be  drawn  through  them 

parallel  to  a  given  straight  line  ? 

15.  Can  you  place  four  points  so  that  it  would  be  impossible  to  draw  a  line 

through  any  two  of  them  parallel  to  a  given  straight  line  ? 

16.  How  many  straight  lines  parallel  to  each  other  can  be  drawn  through  one 

point  ? 

17.  Can  you  draw  more  than  one  set  of  parallel  lines  through  two  points  ? 

18.  What  is  the  greatest  and  what  is  the  least  number  of  parallel  lines  which 

can  be  drawn  through  eight  points  ? 

19.  Place  three  points  so  that  one  straight  line  can  be  drawn  through  them  in 

the  direction  of  northeast. 

20.  Place  the  three  points  so  that  two  straight  lines  can  be  drawn  through  them 

in  the  direction  of  northeast. 

21.  Place  the  points  so  that  no  such  line  can  be  drawn  through  them. 

2.  To  find  the  greatest  number  of  points  of  intersection 
which  can  be  made  by  a  number  of  straight  lines  when 
divided  into  two  groups  in  various  ways,  the  lines  of  one 
group  being  parallel,  and  the  lines  of  the  other  group  cutting 
one  another  at  one  point. 


/ / \ 

/          ~f    ^ 

Suppose  the  number  of  lines  to  \ 

be  six.     We  have  seen  (p.  134)  v    /  ,  \ 

that  six  lines  can  be  divided  into 
two  groups  in  three  ways,  i  and 
5,  2  and  4,  3  and  3.     How  many 
points  of  intersection  can  there 
be  if  one  group  consist  of  paral-          /      i 
lei   lines,  and  the   other  group      /       / 
cut  one  another  at  one  point?  ' 


\ 


140  OBSERVATIONAL    GEOMETRY 

In  this  problem  either  group  in  each  case  can  consist  of  parallel  lines 
or  of  lines  cutting  one  another  at  one  point.  With  six  lines  therefore 
there  are  five  different  arrangements.  In  each  arrangement  there  will 
be  a  group*  of  lines  having  no  point  of  intersection  among  themselves 
since  they  are  parallel,  and  there  will  be  a  group  of  lines  having  one 
point  of  intersection  among  themselves;  and  each  line  of  one  group  will 
cut  each  line  of  the  other  group  at  one  point.  Therefore  the  total  number 
of  points  of  intersection  will  be  found  by  adding  I  to  the  product  of  the 
numbers  of  lines  in  the  two  groups. 

2.  What  is  the  greatest  number  of  points  of  intersection  among  four  straight 

lines  divided  into  two  groups,  one  group  being  parallel,  and  the  other 
group  cutting  one  another  at  one  point  ? 

3.  Find  the  same  for  five  straight  lines. 

4.  Seven  straight  lines. 

5.  Eight  straight  lines. 

6.  If  12  lines  be  divided  into  groups  of  8  and  4,  how  will  the  number  of  points 

of  intersection  when  the  larger  group  is  parallel  compare  with  the  number 
when  the  smaller  group  is  parallel  ? 

7.  Why  is  it  that  when  12  lines  are  divided  into  groups  of  n  and  I,  there  will 

be  one  less  point  of  intersection,  or  one  more,  according  to  the  group 
which  is  made  parallel  ? 

8.  If  the  point  of  intersection  of  the  non-parallel  group  lie  in  the  midst  of  the 

parallel  group,  will  the  total  number  of  points  of  intersection  be  greater 
or  less  than  if  the  point  lie  outside  the  parallels  ? 

9.  What  if  this  point  were  to  lie  on  one  of  the  parallel  lines? 

10.  What  if  a  line  of  one  group  were  parallel  to  a  line  of  the  other  group  ? 

11.  Fifteen  lines,  when  divided  into  two  groups,  the  lines  of  one  group  being 

parallel,  and  the  lines  of  the  other  group  crossing  one  another  at  one  point, 
make  the  following  numbers  of  points  of  intersection, —  14,  15,  27,  37,  45, 
5!»  55>  57  :  what  do  you  notice  about  the  successive  differences  between 
these  numbers  ? 

12.  The  following  is  a  table  of  the  numbers  of  points  of  intersection  made  by 

lines  divided  into  two  groups,  one  group  being  parallel,  and  the  other 
group  crossing  one  another  at  one  point :  — 

3  lines  make  2  or    3  points  of  intersection 

4  "  "  3.  4,  or    5, 

5  "  4,  5,  or    7, 

6  "  "  5,  6,        9,  or  10, 

7  "  "  6,  7,       n,  or  13, 

8  "  "  7,  8,       13,       16,  or  17, 

9  "  "  8,  9,       15,       19,  or  21, 

10    "         "      9,      10,       17,       22,       25,  or  26. 

What  do  you  notice  about  the  increase  in  these  numbers  of  points  when  you 
read  each  column  downwards  ? 

13.  Continue  the  table  for  n  and  12  lines,  using  the  above  scheme  as  a  guide. 


POINTS  OF  INTERSECTION  141 

3.  To  find  the  greatest  number  of  points  of  intersection 
which  straight  lines  can  have  when  divided  into  two  groups 
in  various  ways,  the  lines  of  one  group  being  parallel,  and 
the  lines  of  the  other  group  cutting  one  another  in  the 
greatest  number  of  points. 

1.  Suppose  the  number  of  lines  to  be  six.  These  can  be  divided  into  groups 
of  5  and  i,  4  and  2,  or  3  and  3.  How  many  points  of  intersection  can 
there  be  if  one  group  consist  of  parallel  lines,  and  the  other  group  cut  one 
another  in  the  greatest  number  of  points  ? 


\ 
\ 
\ 
\ 


X 


/     \ 


.'/*  \  /' 


:/\  \ 

I.'     IX  \ 


% 


In  this  problem  either  group  in  each  case  can  consist  of  parallel  lines 
or  of  lines  cutting  one  another  in  the  greatest  number  of  points;  with  six 
lines,  therefore,  there  are  five  arrangements.  In  each  arrangement  there 
will  be  a  group  of  parallel  lines,  having  no  point  of  intersection,  and  a 
group  of  lines  having  the  greatest  number  of  such  points  which  (see 
p.  133)  can  be  found  by  multiplying  the  number  of  lines  in  the  group  by 
i  less  than  that  number  and  dividing  the  product  by  2  ;  also  each  line  in 
one  group  can  cut  each  line  in  the  other  group  at  one  point. 


142  OBSERVATIONAL   GEOMETRY 

Thus  if  the  groups  consist  of  two  parallel  lines  and  four  lines  cutting 
one  another  at  the  greatest  number  of  points,  the  total  number  of  points 

of  intersection  will  be  ^Mp  -f-  8  =  14  points. 

The  number  of  points  for  the  five  arrangements  above  will  be  5,  9,  12, 
14,  and  15. 

2.  What  is  the  greatest  number  of  points  of  intersection  which  four  straight 

lines  can  have  when  divided  in  various  ways  into  two  groups,  the  lines  of 
one  group  being  parallel,  and  those  of  the  other  group  crossing  each  other 
at  the  greatest  number  of  points  ? 

3.  Find  the  same  for  five  straight  lines. 

4.  Find  the  same  for  seven  straight  lines. 

5.  Calculate  the  numbers  of  points  for  twelve  straight  lines,  without  drawing 

diagrams. 

6.  If  twenty  lines  were  divided  into  groups  of   14  and   6,  would   the  total 

number  of  points  of  intersection  be  the  same,  whichever  of  the  two 
groups  were  taken  as  parallel  ? 

7.  In  the  second  diagram  of  the  first  question  what  change  would  there  be  in 

the  answer  if  a  line  of  one  group  were  parallel  to  a  line  of  the  other  group  ? 

8.  In  the  third  diagram  of  the  first  question  what  change  would  there  be  in  the 

answer  if  two  lines  of  the  non-parallel  group  were  to  cut  one  of  the  parallel 
lines  at  the  same  point  ? 

9.  If  a  straight  line  cut  another  straight  line  once,  can  the  two  meet  again  ? 

10.  If  fifteen  straight  lines  be  divided  into  two  groups  in  various  ways,  the  lines 

of  one  group  being  parallel  and  the  lines  of  the  other  group  cutting  each 
other  in  the  greatest  number  of  points,  the  total  numbers  of  points  of  in- 
tersection will  be  as  follows, — 14,  27,  39,  50,  60,  69,  77,  84,  90,  95,  99, 
102,  104,  105:  what  do  you  notice  about  the  successive  differences 
between  these  numbers  ? 

11.  The  following  is  a  table  of  the  total  numbers  of  points  of  intersection  made 

by  lines  divided  into  groups  as  in  the  preceding  question: — 

3  lines  make  2  or    3  points  of  intersection. 

4  "  "  3,         5>°r    6, 

5  "  "  4,        7»        9> or  I0» 

6  "  5,         9,       12,       14,  or  15, 

7  "  "  6,  ii,       15,       18,       20,  or  21, 

8  "  "  7,  13,       18,       22,       25,       27,  or  28, 

9  "  "  8,  15,       21,       26,       30,       33,       35.0*36, 

10    "  9,      17,       24,      30,      35,       39,       42,      44,  or  45. 

What  do  you  notice  about  the  increase  in  these  numbers  of  points  when  you 
read  the  columns  downwards  ? 

12.  Continue  the  table  for  n  and  12  lines,  using  the  above  scheme  as  a  guide. 


POINTS  OF  INTERSECTION 


143 


4.  To  find  the  greatest  number  of  points  of  intersection 
which  straight  lines  can  have  when  divided  in  various  ways 
into  two  groups,  the  lines  of  each  group  cutting  one  another 
at  one  point. 

1.  Suppose  the  number  of  lines  to  be  six.  These  can  be  divided  into  groups 
of  5  and  I,  4  and  2,  or  3  and  3.  How  many  points  of  intersection  can 
there  be  if  the  lines  of  each  group  cut  one  another  in  one  point  ? 


In  this  problem  both  groups  in  each  case  consist  of  lines  cutting  one 
another  at  one  point ;  with  six  lines,  therefore,  there  are  three  different 
arrangements.  In  each  case  there  will  be  one  point  of  intersection  for 
the  lines  of  each  group  among  themselves ;  and  each  line  of  one  group 
will  cut  each  line  of  the  other  group.  The  total  number  of  points  of  in- 
tersection, therefore,  will  be  2  more  than  the  product  of  the  numbers  of 
lines  in  the  two  groups.  Thus  if  the  groups  consist  of  two  and  four  lines, 
the  total  number  of  points  of  intersection  will  be  2  -f-  2  X  4  =  10. 


144  OBSERVATIONAL   GEOMETRY 

2.  What  is  the  greatest  number  of  points  of  intersection  which  four  straight 

lines  can  have  when  divided  in  various  ways  into  two  groups,  the  lines  of 
each  group  cutting  one  another  at  one  point  ? 

3.  Five  straight  lines  ? 

4.  Seven  straight  lines  ? 

5.  Eight  straight  lines  ? 

6.  In  the  second  diagram  of  the  first  question  how  would  the  answer  be  af- 

fected if  a  line  of  one  group  were  parallel  to  a  line  of  the  other  group  ? 

7.  In  the  third  diagram  of  the  first  question  how  would  the  answer  be  affected 

if  the  point  of  intersection  of  one  group  lay  on  a  line  of  the  other  group? 

8.  On  a  map  where  towns  are  represented  by  single  dots,  and  roads  by  single 

lines,  there  are  two  towns,  from  one  of  which  three  straight  roads  lead,  and, 
from  the  other  town,  two  straight  roads.  What  is  the  greatest  possible 
number  of  "cross-roads"  among  them  ? 

9.  In  the  eighth  question  what  difference  would  there  be  if  two  of  the  roads 

were  parallel  ? 

10.  In  the  eighth  question  what  would  the  number  be  if  one  of  the  three  roads 

from  one  town  ran  to  the  other  town? 

11.  If  15  straight  lines  be  divided  into  two  groups  in  various  ways,  the  lines  of 

each  group  cutting  one  another  at  one  point,  the  total  numbers  of  points 
of  intersection  are  as  follows,  — 15,  28,  38,  46,  52,  56,  58:  what  do  you 
notice  about  the  successive  differences  between  these  numbers? 

12.  The  following  is  a  table  of  the  numbers  of  points  of  intersection  made  by 

lines  when  divided  into  groups  as  in  the  preceding  question  :  — 

3  lines  make     3  points  of  intersection. 

4  "  "  4»  or  6, 

5  "  "  5,  or  8, 

6  "  "  6,  10,  or  u, 

7  "  "  7,  12,  or  14, 

8  "  "  8,  14,       17,  or  18, 

9  "  "  9,  1 6,       20,  or  22, 

10    "         "     10,       1 8,       23,       26,  or  27. 

What  do  you  notice  about  the  increase  in  these  numbers  of  points  when  you 
read  the  columns  downwards  ? 

13.  Continue  the  table  for  n  and  12  lines,  using  the  above  scheme  as  a  guide. 


CHAPTER    XXI 

ANGLES 

I.   ANGLES  formed  by  two  straight  lines. 
Draw  two  straight  lines  so  that  they  may  make :  — 

1.  One  angle. 

2.  Two  angles. 

3.  Four  angles. 

4.  Why  cannot  two  straight  lines  form  three  angles  ? 

5.  Why  cannot  two  straight  lines  form  more  than  four  angles  ? 

Draw  two  straight  lines  so  as  to  make :  — 

6.  An  acute  angle. 

7.  A  right  angle. 

8.  An  obtuse  angle. 

9.  Can  you  increase  the  size  of  an  angle  by  lengthening  its  sides? 

10.  If  two  straight  lines  extend  from  a  point,  one  due  east  and  the  other  north- 

west, what  kind  of  an  angle  do  they  form? 

11.  Give  the  table  of  divisions  of  a  right  angle  (see  p.  39). 

With  the  aid  of  a  protractor  draw  two  straight  lines  so  as  to  make  the  fol- 
lowing angles,  and  write  against  each  angle  its  name  —  whether  acute,  right,  or 
obtuse: — 

12.  60°.                        16.   55°.  20.    170°. 

13.  100°.                      17.   140°.  21.   10°. 

14.  20°.                         18.   85°.  22.    150°. 

15.  90°.                         19.   95°.  23.   30°. 
24  How  small  can  an  acute  angle  be  ?  How  great  ? 

25.  How  small  can  an  obtuse  angle  be  ?     How  great  ? 

26.  Is  there  any  variation  in  the  size  of  a  right  angle  ? 

27.  If  an  acute  angle  be  doubled,  can  the  result  be  another  acute  angle  ?     Can 

the  result  be  a  right  angle  ?  Can  the  result  be  an  obtuse  angle  ?  Test 
your  answers  both  by  drawing  diagrams  and  by  giving  the  number  of 
degrees  in  the  angles. 

28.  If  an  obtuse  angle  be  doubled,  what  will  the  result  be  ?     If  a  right  angle  ? 

If  an  acute  angle  ?     Test  your  answers  as  in  the  preceding  question. 


146  OBSERVATIONAL   GEOMETRY 

29.  Draw  two  straight  lines  so  as  to  make  an  angle  of  90°,  and  then  prolong 

one  of  the  lines  through  the  vertex,  thus  forming  another  angle.     How 
great  is  the  second  angle  ? 

30.  Draw  two  straight  lines  so  as  to  make  an  angle  of  60°,  and  then  prolong 

one  of  the  sides  as  before.     With  a  protractor  find  the  size  of  the  second 
angle.     What  is  the  sum  of  the  two  angles  ? 

31.  Proceed  in  the  same  way,  beginning  with  an  angle  of  105°. 

32.  Do  the  same,  with  an  angle  of  45°. 

33.  Do  you  find  that,  allowing  for  errors  in  measurement,  the  sum  of  the  two 

angles  is  the  same  in  each  case  ?     Is  the  sum  180°  ? 

34.  The  supplement  of  an  angle  is  the  difference  between  that  angle  and  two 

right  angles :  are  the  two  angles  in  questions  29-32  each  the  supplement  of 
the  other  ? 

35.  Draw  two  straight  lines  so  as  to  form  at  one  point  angles  of  55°  and  125°. 

36.  150°  and  30°. 

37.  80°  and  100°. 

38.  95°  and  85°. 

39.  If  one  of  two  angles  formed  by  two  straight  lines  be  acute,  what  must  the 

other  be  ? 

40.  Can  the  following  angles  be  formed  at  one  point  by  two  straight  lines,  — 

110°  and  85°  ?     Draw  a  diagram  to  illustrate  your  answer. 

41.  If  one  of  two  angles  formed  at  one  point  by  two  straight  lines  be  83°  20', 

what  is  the  other  ? 

42.  What  is  the  supplement  of  128°  40'  20"  ? 

43.  What  angle  would  be  formed   by  the  halves  of  the  angles  in  the  3oth 

question  ? 

44.  Would  the  answer  to  the  preceding  question  be  the  same  for  the  halves  of 

any  two  angles  whose  sum  is  180°  ? 

45.  The  complement  of  an  angle  is  the  difference  between  that  angle  and  a  right 

angle  :  what  is  the  complement  of  20°  ?  of  82°  ?  of  17°  50"  30"  ? 

46.  Draw  two  straight  lines  so  as  to  make  a  right  angle ;  then  prolong  each 

line  through  the  vertex,  thus  forming  three  more  angles  :  what  is  the  size 
of  these  three  angles  ?     What  is  the  sum  in  degrees  of  all  four  angles  ? 

47.  Draw  two  straight  lines  so  as  to  make  an  angle  of  60°  and  then  prolong  the 

sides  as  in  the  previous  question ;  with  a  protractor  find  the  size  of  each 
of  the  other  angles.     What  is  the  sum  of  the  four  angles  ? 

48.  Proceed  in  the  same  way,  beginning  with  an  angle  of  45°. 

49.  Do  the  same,  with  an  angle  of  105°. 

50.  Do  you  find  that  the  sum  of  the  four  angles  is  the  same  in  each  case  ?     Is 

it  360°  ? 

51.  Are  the  opposite  angles  in  each  case  equal  to  each  other  ? 

52.  Of  how  many  different  sizes  are  the  four  angles  in  any  case  ? 

53.  Is  there  any  one  case  where  the  four  angles  are  of  the  same  size  ? 

54.  Draw  two  straight  lines  so  as  to  form  two  angles  of  80°  and  two  of  100°. 

55.  Draw  two  straight  lines  so  as  to  form  four  angles  as  follows  :     30°,   150°, 

30°,  150°. 

56.  Draw  two  straight  lines  so  as  to  form  four  angles,  one  of  which  is  20°. 


ANGLES  147 

Draw  two  straight  lines  so  as  to  make :  — 

57.  One  right  angle.  61.  One  obtuse  angle. 

58.  Two  right  angles.  62.  One  acute  and  one  obtuse  angle. 

59.  Four  right  angles.  63.  Two  acute  and  two  obtuse  angles. 

60.  One  acute  angle. 

2.  Angles  formed  at  one  point  by  three  straight  lines. 
Draw  three  straight  lines  so  as  to  form  at  one  point  the 

following  angles  *  :  — 

1.  Two  angles.  4.    Five  angles. 

2.  Three  angles.  5.   Six  angles. 

3.  Four  angles. 

Draw  three  straight  lines  so  as  to  form  at  one  point  the  fol- 
lowing groups  of  angles:  — 

6.  i  right  and  i  acute.  15.  2  obtuse  and  2  acute. 

7.  i  obtuse  and  i  acute.  16.  2  right,  i  obtuse,  and  i  acute. 

8.  2  acute.  17.  3  right  and  2  acute. 

9.  i  right  and  2  acute.  18.  2  obtuse  and  3  acute. 
10  i  obtuse  and  2  acute.  19.  i  obtuse  and  4  acute. 

11.  3  acute.  20.  i  obtuse,  i  right,  and  3  acute. 

12.  i  right  and  2  obtuse.  21.  2  right  and  4  acute. 

13.  i  acute  and  2  obtuse.  22.  2  obtuse  and  4  acute. 

14.  3  obtuse.  23.  6  acute. 

3.  Angles  formed  at  two  points  by  three  straight  lines. 
Draw  three  straight  lines  so  as  to  make  at  two  points :  — 

1.  Two  angles.  4.   Five  angles. 

2.  Three  angles.  5.   Six  angles. 

3.  Four  angles.  6.   Eight  angles. 

7.  Why  cannot  three  straight  lines  be  drawn  so  as  to  make  seven  angles  at 

two  points  ? 

Draw  three  straight  lines  so  as  to  make  at  two  points  the 
following  groups  of  angles :  — 

8.  i  right  and  i  acute.  13.   2  obtuse. 

9.  i  right  and  i  obtuse.  14.   3  right. 

10.  i  acute  and  i  obtuse.  15.   2  right  and  i  acute. 

11.  2  right.  16.   2  right  and  i  obtuse. 

12.  2  acute.  17.   2  obtuse  and  r  acute. 

*  It  is  understood  that  the  angles  are  to  be  each  less  than  180°. 


[48 


OBSERVATIONAL    GEOMETRY 


18.  2  acute  and  i  obtuse. 

19.  i  right,  i  acute,  and  i  obtuse. 

20.  4  right. 

21.  2  obtuse  and  2  acute. 

22.  2  right,  i  acute,  and  i  obtuse. 

23.  5  right. 

24.  4  right  and  i  acute. 

25.  4  right  and  i  obtuse. 

26.  i  right,  2  acute,  and  2  obtuse. 


27.  3  acute  and  2  obtuse. 

28.  3  obtuse  and  2  acute. 

29.  6  right. 

30.  4  right,  i  acute,  and  i  obtuse. 

31.  2  right,  2  acute,  and  2  obtuse. 

32.  3  acute  and  3  obtuse. 

33.  Bright. 

34.  4  right,  2  acute,  and  2  obtuse. 

35.  4  acute  and  4  obtuse. 


4.    Angles  formed  at  three  points  by  three  straight  lines. 
Draw  three  straight  lines  so  as  to  make  at  three  points : 

1.  Three  angles.  4.   Six  angles  7.  Nine  angles. 

2.  Four  angles.  5.   Seven  angles.  8.  Ten  angles. 

3.  Five  angles.  6.  Eight  angles.  9.  Twelve  angles. 
10.  Why  cannot  eleven  angles  be  formed  in  this  way? 

Draw  three  straight  lines  so  as  to  form  at  three  points  the 
following  groups  of  angles  :  — 


11.  .3  acute. 

12.  i  right  and  2  acute. 

13.  2  acute  and  i  obtuse. 

14.  2  right  and  2  acute. 

15.  i  right,  2  acute,  and  i  obtuse. 

16.  3  acute  and  i  obtuse. 

17.  2  acute  and  2  obtuse. 

18.  2  right,  2  acute,  and  i  obtuse. 

19.  i  right,  2  acute,  and  2  obtuse. 

20.  3  acute  and  2  obtuse. 

21.  3  obtuse  and  2  acute. 

22.  4  right  and  2  acute. 

23.  2  right,  2  acute,  and  2  obtuse. 

24.  i  right,  3  acute,  and  2  obtuse. 

25.  4  acute  and  2  obtuse. 

26.  3  acute  and  3  obtuse. 

27.  4  right,  2  acute,  and  i  obtuse. 


28.  2  right,  3  acute,  and  2  obtuse. 

29.  i  right,  3  acute,  and  3  obtuse. 

30.  4  acute  and  3  obtuse. 

31.  3  acute  and  4  obtuse. 

32.  4  right,  2  acute,  and  2  obtuse. 

33.  2  right,  3  acute,  and  3  obtuse. 

34.  4  acute  and  4  obtuse. 

35    4  right,  3  acute,  and  2  obtuse. 

36.  i  right,  4  acute,  and  4  obtuse. 

37.  5  acute  and  4  obtuse. 

38.  4  acute  and  5  obtuse. 

39.  4  right,  3  acute,  and  3  obtuse. 

40.  2  right,  4  acute,  and  4  obtuse. 

41.  5  acute  and  5  obtuse. 

42.  4  right,  4  acute,  and  4  obtuse. 

43.  6  acute  and  6  obtuse. 


CHAPTER    XXII 


TRIANGLES,   QUADRILATERALS,   AND   POLYGONS 

I.    REVIEW  what  is  said  about  triangles  on  pp.  34-36. 

1.   Construct  a  triangle  having  one  side  3  cm.  long,  and  the  angles  at  the  end 
of  that  side  60°  and  45°. 


3  cm. 


Draw  a  straight  line  AB  3  cm.  long.  At  A  draw  a  line  so  as  to  make 
with  AB  an  angle  of  45°;  and  at  B  draw  a  line  so  as  to  make  with  AB 
an  angle  of  60°;  prolong  these  lines  until  they  meet  at  C;  then  ABC  will 
be  the  triangle. 

Measure  with  the  protractor  the  angle  C. 

What  is  the  sum  of  the  angles  A,  B,  and  C? 

2.  Construct  a  triangle  having  one  side  5  cm.  long,  and  the  angles  at  the  end 

of  that  side  30°  and  50°. 
Measure  the  third  angle,  and  find  the  sum  of  all  three  angles. 

3.  Do  the  same,  taking  for  the  side  4  cm.,  and  for  the  angles,  120°  and  40°. 

4.  Do  the  same,  taking  for  the  side  4  cm.,  and  for  the  angles,  20°  and  40°. 

5.  Take  the  side  5  cm.,  and  the  angles  70°  and  20°. 

Allowing  for  inaccuracy  in  measuring  the  angles,  do  you  find  that  these  five 
triangles  agree  in  the  sum  of  their  angles  ?  Is  the  sum  iSo°? 

6.  Construct  a  triangle  having  one  side  4  cm.  and  the  angles  at  the  ends  each 

40°.     Measure  the  third  angle,  find  the  sum  of  all  three  angles,  and  com- 
pare the  lengths  of  the  sides  opposite  the  equal  angles. 

7.  Do  the  same,  taking  for  the  side  5  cm.,  and  for  the  equal  angles  30°. 

8.  Do  the  same,  taking  for  the  side  5  cm.,  and  for  the  equal  angles  45°.     From 

the  last  three  triangles,  what  do  you  find  about  the  equality  of  sides  when 
there  are  two  equal  angles  in  a  triangle  ? 


1 50  OBSERVATIONAL  GEOMETRY 

9.  Construct  a  triangle  having  a  side  5  cm.  long,  and  the  angles  at  the  end 
each  60°.  What  do  you  find  about  the  three  angles  and  the  three  sides  of 
this  triangle  ? 

10.  Construct  a  triangle  having  a  side  8  cm.  long,  and  the  angles  at  the  ends, 

30°  and  60°.     Measure  the  third  angle  and  the  other  two  sides. 
(u)    Does  the  longest  side  lie  opposite  the  greatest  angle  ? 

(b)  Does  the  shortest  side  lie  opposite  the  smallest  angle  ? 

(c )  60°  is  twice  30° ;  but  is  the  side  opposite  60°  twice  as  long  as  the  side 

opposite  30°  ? 

(d)  Is  there  any  side  twice  as  long  as  the  side  opposite  30°  ? 

11.  What  is  the  sum  of  the  angles  of  any  triangle  ? 

12.  If  the  three  angles  are  equal,  how  many  degrees  are  there  in  each? 

13.  How  many  of  the  angles  of  a  triangle  can  be  obtuse  ? 

14.  How  many  of  the  angles  can  be  right  angles  ? 

15.  Construct  a  triangle  having  three  acute  angles. 

16.  Construct  a  triangle  having  one  obtuse  and  two  acute  angles. 

17.  Construct  a  triangle  having  one  right  and  two  acute  angles. 


QUADRILATERALS 

2.   Review  what  is  said  about  quadrilaterals  on  pp.  20-22. 
Construct  quadrilaterals  whose  angles  shall  be  as  follows: 

1.  4  right. 

2.  2  right,  i  acute,  and  I  obtuse. 

3.  i  right,  2  acute,  and  i  obtuse. 

4.  i  right,  i  acute,  and  2  obtuse. 

5.  3  acute  and  i  obtuse. 
6  2  acute  and  2  obtuse. 

7.  i  acute  and  3  obtuse. 

8.  90°,  90°,  90°,  90°.     Let  all  the  sides  be  equal.     What  is  the  name  of  this 

figure  ? 

9.  go0,  90°,  90°,  90°.     Make  a  figure  which  shall  not  have  all  its  sides  equal. 

Notice  whether  any  of  the  sides  are  equal  or  parallel.     What  is  the  name 
of  this  figure  ? 

10.  90°,  90°,  160°,  20°.     Let  the  figure  have  two  parallel  sides.     What  is  its 

name  ? 

11.  90°,  90°,  160°,  20°.    Let  the  figure  have  no  parallel  sides.  What  is  its  name  ? 

12.  100°,  80°,  1 00°,  80°.     Arrange  the  angles  so  that  the  figure  may  be  a 

parallelogram. 

13.  Arrange  the  angles  of  the  preceding  problem  so  that  the  figure  may  be  a 

trapezoid. 

14.  150°,  30°,  150°,  30°.     Let  the  figure  be  a  parallelogram. 

15.  Change  the  figure  of  the  previous  problem  into  a  rhombus. 

16.  What  is  the  difference  between  a  rhombus  and  a  parallelogram  ? 


POLYGONS  151 

17.  Could  the  sides  of  a  rhombus  and  those  of  a  parallelogram  be  equal,  each 

to  each  ? 

18.  What  is  the  difference  between  a  rectangle  and  a  parallelogram? 

19.  Could  the  sides  of  a  rectangle  be  equal,  each  to  each,  to  those  of  a  square  ? 

20.  What  is  the  difference  between  a  square  and  a  rectangle  ? 

21.  What  is  the  difference  between  a  rhombus  and  a  square  ? 

22.  Could  the  sides  of  a  rhombus  and  those  of  a  square  be  equal,  each  to  each  I 

23.  In  what  particular  respect  do  the  square  and  rectangle  agree? 

24.  In  what  do  the  rhombus  and  square  agree  ? 

25.  What  can  be  said  alike  of  all  four  figures,  —  rhombus,  square,  rectangle,  and 

parallelogram  ? 

POLYGONS 

3.  Review  what  is  said  about  polygons  on  pp.  72-78. 
How  many  sides  have  the  following  polygons?  — 

1.  Quadrilateral.  6.  Nonagon. 

2.  Pentagon.  7.  Decagon. 

3.  Hexagon.  8.  Dodecagon. 

4.  Heptagon.  9.  Pentedecagon. 

5.  Octagon.  10.  Icosagon. 

The  angles  of  a  polygon  are  the  angles  made  by  its  sides 
meeting,  as  ABC,  BCD,  etc. 


C  D 

They  are  measured  within  the  polygon,  and  are  sometimes 
called  interior  angles. 

How  many  angles  have  the  following  polygons  :  — 

11.  Quadrilateral.  14.    Heptagon. 

12.  Pentagon.  15.    Octagon. 

13.  Hexagon.  16.   Nonagon. 

17.  How  does  the  number  of  angles  of  a  polygon  compare  with  the  number  of 

sides  ? 

18.  How  many  angles  has  a  dodecagon  ? 

19.  An  icosagon  ? 

20.  A  polygon  of  thirty  sides  ? 


152  OBSERVATIONAL   GEOMETRY 

The  vertices  of  a  polygon  are  the  vertices  of  its  angles,  as 
A,  B,  C,  etc. 

How  many  vertices  have  the  following  polygons? 

21.  Rhombus.  25.  Octagon. 

22.  Pentagon.  26.  Nonagon. 

23.  Hexagon.  27.  Decagon. 

24.  Heptagon.  28.  Dodecagon. 

29.  How  does  the  number  of  vertices  compare  with  the  number  of  angles  of  a 

polygon  ?     With  the  number  of  sides  ? 

30.  How  many  vertices  has  a  polygon  of  forty  sides? 


A  diagonal  of  a  polygon  is  a  straight  line  which  joins  any 
two  vertices,  except  those  already  connected  by  the  sides, 
as  AC,  AD,  etc.  If  all  possible  diagonals  be  drawn  from  any 
one  vertex,  as  A,  the  polygon  will  be  divided  into  triangles, 
ABC,  A  CD,  etc. 

Into  how  many  triangles  can  the  following  polygons  be 
divided  by  drawing  diagonals  from  any  vertex :  — 

31.  Quadrilateral.  35.  Octagon. 

32.  Pentagon.  36.  Nonagon. 

33.  Hexagon.  37.  Decagon. 

34.  Heptagon.  38.  Triangle. 

39.  The  number  of  triangles  is  always  less  than  the  number  of  sides  by  the 

same  amount :  how  much  less  ?   why  ? 

40.  Into  how  many  triangles  can  a  polygon  of  thirty  sides  be  divided  by  draw- 

ing diagonals  from  any.  vertex  ? 

Draw  all  possible  diagonals  in  the  following  polygons,  and 
find  the  number  in  each  case. 

41.  Quadrilateral.  43.   Heptagon. 

42.  Hexagon.  44.    Octagon. 


POLYGONS 


'53 


45.  The  number  of  diagonals  which  can  be  drawn  from  any  one  vertex  is  always 

less  than  the  number  of  sides  by  the  same  amount :  how  much  less  ?  why  ? 

46.  If  you  were  to  multiply  the  number  of  diagonals  from  one  vertex  by  the 

number  of   vertices,   the  product  would  be  greater  than  the  number  of 
different  diagonals :  how  many  times  greater  ? 

47.  What  rule  can  you  give  for  finding  the  total  number  of  different  diagonals 

in  any  polygon  ? 

48.  Calculate  the  total  number  of  diagonals  in  a  polygon  of  twenty  sides. 

49.  Calculate  the  same  for  a  polygon  of  thirty  sides. 

50.  Calculate  it  for  a  polygon  of  forty-eight  sides. 


4.   To  find  the  sum  of  all  the  angles  of  a  polygon. 
ABCDEF\s  a  polygon  of  six  sides. 

From  Ay  one  of  the  vertices,  draw  all  the  diagonals,  thus 
dividing  the  polygon  into  triangles. 

1.  How  does  the  number  of  triangles  compare  with  the  number  of  sides  ? 

2.  Can  you  see  that  the  sum  of  the  angles  of  the  triangles  is  the  same  as  the 

sum  of  the  angles  of  the  polygon  ? 

3.  What  is  the  sum  of  the  angles  of  any  triangle  ? 

4.  What  is  the  sum  of  the  angles  of  all  the  triangles,  ABC,  A  CD,  etc.,  together? 

5.  What,  then,  is  the  sum  of  all  the  angles  of  the  polygon  ABCDEF? 

6.  If  the  sum  is  eight  right  angles,  what  is  the  sum  in  degrees  ? 


154  OBSERVATIONAL   GEOMETRY 

7.   Complete  the  following  table,  showing  the  number  of  triangles  which  com- 
pose the  polygons,  and  the  sum  of  their  angles  :  — 

3  sides,  i  triangle,    2  right  angles. 

4  "       2  triangles,  4          " 

5  " 

6  "  "  " 


What  do  you  notice  about  the  increase  of  the  numbers  when  you  read  the 
columns  downwards? 

From  the  preceding  results  a  rule  can  be  derived  :  — 

To  find  the  sum  of  the  angles  of  any  polygon,  subtract  2 

from  the    number   of  sides  and  double  the  remainder:   the 

result  will  be  the  sum  of  the  angles  expressed  in  right  angles; 

the   result  multiplied  by  90  will  be  the  sum  of  the  angles 

expressed  in  degrees. 

Calculate  the  sum  of  the  angles  of  the  following  polygons, 

expressing  the  results  both  in  right  angles  and  in  degrees: 

8.  Octagon.  15.    Twenty-four  sides. 

9.  Nonagon.  16.   Twenty-five  sides. 

10.  Decagon.  17.  Thirty  sides. 

11.  Dodecagon.  18.  Thirty-two  fcides. 

12.  Pentedecagon.  19.  Forty  sides. 

13.  Icosagon.  20.  Forty-eight  sides. 

14.  Eighteen  sides. 

Since  the  angles  of  a  regular  polygon  are  equal,  the  size 
of  one  of  the  angles  can  be  found  by  dividing  the  sum  of  all 
the  angles  by  the  number  of  sides  of  the  polygon. 

Find  in  degrees  the  size  of  one  angle  of  the  following 
regular  polygons  :  — 

21.  Pentagon.  26.  Decagon. 

22.  Hexagon.  27.  Dodecagon. 

23.  Heptagon.  28.  Pentedecagon. 

24.  Octagon.  29.  Icosagon. 

25.  Nonagon.  30.  Thirty-two  sides. 

5.   Pentagons  and  Hexagons.     Pentagons  are  possible  with 
ten  different  combinations  of  obtuse,  right,  and  acute  angles. 


POLYGONS  155 

Particular  care  should  be  taken  in  their  construction  to  pro- 
duce figures  which  shall  be  distinct  in  the  sizes  of  the  angles 
as  well  as  neat. 

Construct  pentagons  which  shall  have  the  following 
angles :  — 

31.  5  obtuse.  36.  3  obtuse,  2  acute. 

32.  4  obtuse,  I  right.  37.  2  obtuse,  3  right. 

33.  4  obtuse,  I  acute.  38.  2  obtuse,  2  right,  i  acute. 

34.  3  obtuse,  2  right.  39.  2  obtuse,  I  right,  2  acute. 

35.  3  obtuse,  i  right,  i  acute.  40.  2  obtuse,  3  acute. 

Hexagons  are  possible  with  ten  different  combinations  of 
obtuse,  right,  and  acute  angles. 

Here,  also,  particular  care  should  be  taken  to  produce  dis- 
tinct, neat,  symmetrical  figures. 

Construct  hexagons  which  shall  have  the  following 
angles :  — 

41.  6  obtuse.  46.  4  obtuse,  i  right,  i  acute. 

42.  5  obtuse,  i  right.  47.  3  obtuse,  2  right,  i  acute. 

43.  5  obtuse,  i  acute.  48.  3  obtuse,  i  right,  2  acute. 

44.  4  obtuse,  2  right.  49.  3  obtuse,  3  right. 

45.  4  obtuse,  2  acute.  50.  3  obtuse,  3  acute. 


CHAPTER  XXIII 

CIRCLES 

I .   Positions  of  Circles  with  Reference  to  one  another.    Review 
what  is  said  about  circles  on  pp.  87-96. 

I.  Two  circles  may  have  the  same  centre,  in  which  case 
they  are  called  concentric  circles  and  their  circumferences  have 
no  points  in  common. 


2.  •  Two  circles  may  have  different  centres.  Draw  diagrams 
to  illustrate  the  following  cases :  — 

(a)  One  circle  lying  wholly  within  the  other  with  the  circumferences  having 
no  point  in  common. 

(ft)  One  circle  lying  wholly  within  the  other  with  the  circumferences  having 
one  point  in  common. 

(r)  One  circle  lying  partly  within  the  other,  the  circumferences  having  two 
points  in  common. 

((f)  The  circles  lying  wholly  outside  one  another  with  the  circumferences 
having  one  point  in  common. 

(<?)  The  circles  lying  wholly  outside  one  another  \vith  the  circumferences 
having  no  point  in  common. 

1.  Draw  two  concentric  circles  so  that  the  radius  of  one  may  be  equal  to  a 

diameter  of  the  other. 

2.  Draw  two  circles  so  that  the  centre  of  each  may  lie  on  the  circumference  of 

the  other. 


CIRCLES 


'57 


Draw  two  circles  with  their  centres  at  the  ends  of  a  straight  line  AB,  so 
that  — 

3.  Their  areas  may  have  no  point  in  common. 

4.  Their  areas  may  have  one  point  in  common. 

5.  The  area  of  one  may  be  included  in  the  area  of  the  other. 

6.  Draw  a  circle ;  then  draw  two  more  circles  inside  the  first,  each  having  a 

diameter  equal  to  a  radius  of  the  first  circle. 

7.  Draw  three  concentric  circles  so  that  the  radius  of  the  greatest  may  be 

equal  to  the  sum  of  the  radii  of  the  other  two. 

Draw  two  circles  with  their  centres  at  the  ends  of  a  straight  line  AB,  so  that 
their  circumferences  may  have  — 

8.  No  point  in  common. 

9.  One  point  in  common. 
10.   Two  points  in  common. 

Draw  two  circles  so  that  the  distance  between  their  centres  may  be- 
ll.  Equal  to  the  sum  of  their  radii. 
Less  than  the  sum  of  their  radii, 
o. 


12. 
13. 

14.  Greater  than  the  difference  between  their  radii. 

15.  Equal  to  the  difference  between  their  radii. 
Less  than  the  difference  between  their  radii. 


16. 
17. 


18. 


19. 


20. 


Draw  three  circles  of  equal  radii,  with  their  centres  in  a  straight  line,  so 

that  the  circumference  of  the  middle  circle  may  pass  through  the  centres 

of  the  other  two. 
Draw  three  unequal  circles,  two  inside  the  third,  with  their  centres  in  a 

straight  line,  so  that  the  radius  of  one  may  be  equal  to  the  sum  of  the  radii 

of  the  other  two. 
Draw  three  equal  circles  with  their  centres  in  a  straight  line  which  is  equal 

to  the  sum  of  their  diameters. 
Draw  three  circles  so  that  the  centres  of  two  may  each  lie  in  two  of  the 

other  circumferences. 


2.  Chords  of  Circles.  A  chord  is  a  straight  line  which  con- 
nects the  ends  of  an  arc.  The  chord  is  said  to  subtend  the 
arc.  The  word  originally  meant  the  string  of  a  musical  instru- 


iS8 


OBSERVATIONAL  GEOMETRY 


ment  such  as  a  harp ;   it  is  the  same  as  cord,  but  has  a  differ- 
ent spelling  in  geometry. 

1.  Draw  a  chord  which  shall  be  equal  to  a  radius  of  the  circle. 

2.  Draw  a  chord  through  the  centre  of  the  circle.     What  special  name  do  you 

give  to  this  chord  ? 

3.  Draw  the  longest  chord  you  can  in  a  circle.    What  do  you  notice  about  this 

chord  ? 

4.  Draw  two  unequal  chords  perpendicular  to  each  other. 

5.  Draw  a  chord  of  any  length.     Draw  diameters  through  its  ends,  and  three 

other  chords  through  the  ends  of  the  diameters.     What  shape  has  the 
quadrilateral  formed  by  the  four  chords  ? 

6.  If  AB  is  the  diameter  of  a  circle,  where  is  the  centre  ? 

7.  Draw  a  diameter.     Then  draw  four  chords  of  various  lengths,  each  perpen- 

dicular to  this  diameter  ?     What  do  you  notice  about  the  parts  into  which 
the  diameter  divides  each  chord  ? 

8.  If  you  were  to  draw  a  perpendicular  at  the  middle  of  a  chord,  through  what 

particular  point  of  the  circle  would  it  pass  ? 

9.  The  last  two  questions  suggest   methods  of  finding  the  centre  of  a  circle 

when  the  centre  is  not  indicated  in  the  figure. 
Do  you  see  how  it  could  be  done  ? 
10.   Through  one  point  in  a  circumference  how  many  chords  of  the  same  length 

can  you  draw  ? 
How  many  diameters  ? 

3.  Subdivision  of  a  Circumference  into  Arcs.  AB  and  CD 
are  diameters  drawn  perpendicular  to  each  other.  You  will 
notice  that  they  divide  the  circumference  into  four  equal  arcs. 
Also,  if  radii  be  drawn  dividing  the  right  angle  BOC  into 
four  equal  angles,  the  arc  BC  will  be  divided  into  four  equal 
arcs,  each  arc  corresponding  to  one  of  the  four  angles. 


Just  as  the  right  angle  BOC  can  be   divided  into  ninety 
equal  parts,  each  of  i°,  so  can  the  arc  BC  be  divided  into 


CIRCLES  159 

ninety  equal  parts,  each  called  an  arc  of  i°;  and  the  arc  of 
i°  is  subdivided  into  arcs  of  i'  and  ir/.  The  whole  circum- 
ference, therefore,  consists  of  360  parts,  each  of  which  is  an 
arc  of  1°. 

This   idea  is   expressed  by  saying  that  "  an  angle  at  the 
centre   is   measured  by  the  arc  between   its  sides ;  "  which 


means  that  an  angle  formed  by  two  radii  is  the  same  part  of 
four  right  angles  as  the  arc  between  the  ends  of  the  radii  is 
of  the  whole  circumference.  Thus,  if  the  angle  AOB  is  40°, 
the  arc  AB  is  also  40°. 

To  mark  off  an  arc  of  required  size  upon  the  circumference 
of  a  given  circle. 

1st.  With  the  aid  of  a  protractor. 

B 


Let  O  be  the  centre  of  the   given  circle  and  70°  be  the 
required  arc. 

Draw  OA  and  OB,  radii  forming  an  angle  of  70°. 

Then  AB  will  be  the  required  arc. 

Draw  circles  with  any  convenient  radii,  and  mark  off  the 
following  arcs,  one  on  each  circumference,  with  the  aid  of  a 
protractor :  — 
1.   20°.  2.    50°.  3.  80°.  4.    140°.  5.  1 60°. 


i6o 


OBSERVATIONAL    GEOMETRY 


An  arc  of  any  required  size  can  be  constructed  without 
drawing  the  whole  circumference. 

If  the  required  arc  be  55°,  construct  an  angle  XOYof  55°. 
Then  with  the  vertex  0  as  a  centre,  and  a  radius  equal  to  the 
radius  of  the  circle,  draw  the  arc  AB  between  the  sides  of  the 
angle.  This  will  be  the  required  arc. 


Construct  the  following  arcs  without  drawing  more  of  the 
circumference  in  any  case:  — 

11.     120°. 


10. 


6.   40°.        7.   65°.        8.    100°.        9.    115°. 

2d.    With  the  aid  of  compasses. 

Certain  arcs  can  be  constructed  more  rapidly  and  neatly 
with  the  aid  of  compasses  than  with  the  aid  of  a  protractor. 
Some  of  the  principal  cases  are  the  following:  — 
(a)     To  construct  an  arc  of  90°. 


Draw  two  diameters,  AB  and  CD,  perpendicular  to  each  other.  Then 
any  one  of  the  four  arcs  thus  marked  off  will  be  the  required  arc  of  90°; 
for  each  is  one-fourth  of  the  whole  circumference. 


CIRCLES 


161 


(b)     To  construct  an  arc  of  60°. 

B 


Draw  the  chord  AB  equal  to  the  radius  of  the  circle.  Then  the  arc 
AB  is  the  required  arc  of  60°. 

For  if  you  draw  the  radii  OA  and  OB,  the  triangle  A  OB  will  be  equi- 
lateral, each  side  being  equal  to  a  radius  ;  each  angle,  therefore,  is  equal 
to  60° ;  and  if  the  angle  O  is  60°,  its  corresponding  arc  AB  is  also  60°. 


(<:)     To  construct  an  arc  of  1 50' 


First  mark  off  an  arc  AB  equal  to  90°. 
Then,  beginning  at  B  mark  off  an  arc  BC  equal  to  6oc 
will  be  the  required  arc  of  1 50°. 
For  A  C  =  AB  +  BC  =  90°  -f  60°  ==  1 50°. 

To  construct  an  arc  of  30°. 


The  arc  AC 


First  mark  off  an  arc  AB  equal  to  90°. 
Then  mark  off  a  part  of  AB,  namely,  AC,  equal  to  60°. 
be  the  required  arc  of  30°. 

For  CB  —  AB  —  A  C  =  90°  —  60°  =  30°. 


CB  will 


162 


OBSERVATIONAL   GEOMETRY 


To  construct  an  arc  of  45°. 

C 


First  mark  off  an  arc  AB  equal  to  90°,  and  draw  its  chord.  Then 
draw  the  radius  OC  through  the  middle  point  M  of  the  chord  AB.  The 
arcs  AC  and  CB  will  each  be  equal  to  the  required  arc  of  45°.  Fora 
radius  (or  diameter)  which  passes  through  the  middle  of  a  chord  will  also 
pass  through  the  middle  of  the  arc  subtended  by  the  chord. 

Thus  A  C  =  CB  =  one-half  of  90°  =  45°. 

With  the  aid  of  compasses  construct  the  following  arcs : 


11.  15°. 
16.  7°  30'. 


12.  75°.     . 
17.  37°  30'- 


13.    105°. 
18.   52°  30'. 


14.    120°. 
19.  Q703 


15.   135°. 
20.   67°  30'. 


4.   Tangents.     A  tangent  is  a  straight  line  which  has  only 
one  point  in  common  with  a  circumference,  however  far  the 
line  may  be  prolonged. 

The  word  tangent  means  "  touching." 

Besides  the  definition  there  are  two  facts 
to  be  noticed  about  a  tangent. 

ist.  A  tangent  has  the  same  direction  as 
the  circumference  at  the  point  of  contact. 

The  curves  of  railroads  depend  on  this  fact,  as  was  explained  on  p.  92 ;  the 
straight  rails  are  tangent  to  the  curved  ones  at  the  points  where  they  meet. 

2d.    A  tangent  is  perpendicular  to  the 
radius  (or  diameter)  drawn  to  the  point   A 
of  contact. 

This  affords  a  method  of  drawing  a  tangent, 
wnen  you  know  the  point  of  tangency. 

Suppose  that  you  wish  to  draw  a  tangent  at 
the  point  P.  First  draw  the  radius  OP.  Then, 


CIRCLES 


163 


at  P  draw  the  straight  line  AB  perpendicular  to  OP.     AB  will  be  the 
required  tangent. 

1.  Draw  a  circle  ;  take  any  point  in  the  circumference  and  construct  a  tangent 

at  that  point. 

2.  In  the  annexed  figure,  AB  is  not  a  tangent  to  the  circle :  why  not? 


3.  Draw  a  tangent  at  each  end  of  some  diameter,  and  compare  their  directions. 

4.  Draw  two  diameters  perpendicular  to  each  other,  and  then  draw  a  tangent 

at  each  end  of  the  diameters ;  prolong  the  tangents  until  they  meet.  What 
shape  has  the  figure  formed  by  the  tangents  ? 

5.  Find  three  points  on  a  circumference,  so  situated  that  the  three  arcs  into 

which  the  circumference  is  divided  may  each  be  120°.  Then  draw  a  tangent 
at  each  point,  and  prolong  the  three  until  they  meet.  What  shape  has  the 
figure  formed  by  the  tangents  ? 

Two  circles  are  said  to  be  tangent  to  each  other  when  they  can  be 
tangent  to  the  same  straight  line  at  the  same  point. 


6.  In  the  preceding  diagram  the  circles  are  called  tangent  externally,  because 

one  circle  lies  outside  the  other.     How  does  the  distance  between  their 
centres  compare  with  the  two  radii? 

7.  Draw  a  diagram  showing  two  circles  tangent  internally,  —  that  is,  one  circle 

lying  within  the  other.     How  does  the  distance  between  their  centres  com- 
pare with  the  two  radii  ? 

8.  Draw  a  diagram  showing  three  circles  all  tangent  at  the  same  point.    Are 

the  three  centres  and  the  point  of  contact  all  in  the  same  straight  line? 

9.  How  would  you  draw  a  line  through  a  point  which  lies  in  a  given  circum- 

ference, so  that  it  would  contain  the  centres  of  all  circles  which  could  be 
tangent  to  the  given  circle  at  that  point  ? 


1 64  OBSERVATIONAL   GEOMETRY 

10.  If  a  circle  has  a  radius  of  14  mm.,  how  would  you  draw  a  line  which  would 
contain  the  centres  of  all  circles  which  have  a  radius  of  8  mm.  and  are 
tangent  externally  to  the  first  circle  ? 

5.    Secants.     A  secant  is  a  straight  line  which  cuts  a  cir- 
cumference in  two  points,  as  AB. 
The  word  secant  means  "  cutting." 


If  a  line  passes  beyond  the  circumference  at  only  one  point, 
it  is  still  considered  to  be  a  secant.  In  fact,  the  length  of  a 
secant  in  many  problems  is  understood  to  be  the  distance 
from  the  point  outside  the  circle,  where  it  begins,  up  to  the 
second  point  where  it  reaches  the  circumference. 

1.  If  you  prolong  a  chord,  what  does  it  become  ? 

2.  Draw  a  chord  of  a  circle  and  a  secant  whose  length  is  equal  to  that  of  the 

chord. 

3.  Why  cannot  a  tangent  be  changed  into  a  secant  by  prolonging  it? 

4.  From  a  point  outside  a  circle  draw  four  secants  each  ending  where  it  reaches 

the  circumference  the  second  time.     Are  those  secants  longer  or  shorter 
which  pass  near  the  centre  of  the  circle  ? 

5.  How  would  you  draw  the  longest  possible  secant  from  a  point  outside  a 

circle  ? 

6.  From  a  point  outside  a  circle  how  would  you  draw  a  secant  ending  at  the 

second  point  where  it  meets  the  circumference,  so  that  the  least  possible 
part  may  lie  outside  the  circle  ? 

7.  In  two  concentric  circles  draw  a  line  which  shall  be  a  chord  of  one  and  a 

secant  of  the  other. 

8.  In  two  concentric  circles  draw  a  line  which  shall  be  a  secant  of  one  and  the 

longest  possible  chord  of  the  other. 

9.  In  two  intersecting   circles  draw  a  line  which  shall  be  a  chord  of  both. 

Then  change  this  common  chord  into  a  common  secant  having  twice  the 
length  of  the  chord. 

10.  Draw  two  circles  which  shall  be  tangent  externally.  Then  draw  a  line  with 
both  ends  in  the  circumferences,  so  that  it  may  be  a  secant  of  each  circle 
and  equal  to  the  sum  of  their  diameters. 


CHAPTER    XXIV 

REGULAR    POLYGONS 

I.    A  regular  polygon  is  a  polygon  which  is  both  equi- 
lateral and  equiangular.     See  p.  73. 

D 


A  B 

A  Regular  Polygon 

The  construction  of  regular  polygons  is  most  easily  effected 
with  the  aid  of  circles  and  depends  on  the  following  truths: 

1st.  If  a  circumference  be  divided  into  equal  arcs,  the 
chords  of  those  arcs  will  be  equal,  and  the  angles  formed  by 


Inscribed  Regular  Polygon 

the  chords  will  also  be  equal ;  the  polygon,  therefore,  thus 
formed  will  be  regular.  The  polygon  is  then  said  to  be 
inscribed  in  the  circle. 


1 66  OBSERVATIONAL   GEOMETRY 

Any  polygon,  whether  regular  or  not,  is  called  an  inscribed  polygon  when  its 
sides  are  all  chords  of  a  circle. 

2d.  If  a  circumference  be  divided  into  equal  arcs,  the 
tangents  drawn  at  the  points  of  division  of  the  arcs,  and  pro- 
longed until  they  meet  one  another,  will  be  equal,  and  the 


Circumscribed  Regular  Polygon 

angles  formed  by  the  tangents  will  also  be  equal;  the  poly- 
gon, therefore,  thus  formed  will  be  regular.  The  polygon  is 
then  said  to  be  circumscribed  about  the  circle. 

Any  polygon,  whether  regular  or  not,  is  called  a  circumscribed  polygon  when 
its  sides  are  all  tangents  to  a  circle. 

To  construct  a  regular  polygon,  therefore,  of  any  number 
of  sides ;  first  draw  a  circle  and  divide  the  circumference  into 
as  many  equal  parts  as  the  polygon  is  to  have  sides,  with 
the  aid  of  compasses  or  protractor  as  shown  on  p.  159; 
then  at  the  points  of  division  of  the  arcs  draw  chords  or 
tangents,  according  as  the  polygon  is  to  be  inscribed  or 
circumscribed. 

Construct  regular  inscribed  and  circumscribed  polygons 
of  the  following  numbers  of  sides,  using  one  circle  for  two 
polygons  of  the  same  number  of  sides:  — 

1.  Triangle,  inscribed.  6.  Pentagon,  circumscribed. 

2.  Triangle,  circumscribed.  7.  Hexagon,  inscribed. 

3.  Square,  inscribed.  8.  Hexagon,  circumscribed. 

4.  Square,  circumscribed.  9.  Octagon,  inscribed. 

5.  Pentagon,  inscribed.  10.  Octagon,  circumscribed. 


REGULAR  POLYGONS  167 


The  centre  of  a  regular  polygon  is  the  same  point  as  the 
centre  of  its  inscribed  and  circumscribed  circles. 

An  angle  at  the  centre  of  a  regular  polygon  is  the  angle 
formed  by  two  lines  drawn  from  the  centre  of  the  polygon 
to  two  consecutive  vertices.  In  any  regular  polygon  this 
angle  is  equal  to  360°  divided  by  the  number  of  sides  of  the 
polygon. 

Find  in  degrees  the  size  of  an  angle  at  the  centre  of  the 
following  regular  polygons :  — 

1.   Triangle.    2.    Square.         3.    Pentagon.    4.    Hexagon.  5.    Heptagon. 

6.    Octagon.     7.   Nonagon.     8.    Decagon.     9.    Pentedecagon.    10.    Icosagon. 

2.  To  find  the  length  of  the  circumference  of  a  circle. 
•  The  length  of  a  curved  line  is  usually  difficult  to  find  by 
actual  measurement.  Sometimes  you  can  apply  a  flexible 
ruler,  such  as  a  tape,  which  will  bend  so  as  to  follow  the 
curve.  Usually,  however,  the  length  of  a  curve  is  found  by 
calculations  which  depend  on  the  nature  of  the  particular 
curve  in  question.  For  that  reason  engineers  and  makers 
of  machinery  are  careful  to  use  curves  whose  nature  is 
known. 

The  circumference  of  a  circle  is  one  of  the  curves  whose 
length  is  most  easily  calculated.  Geometers  have  proved 
that  the  length  of  a  circumference  is  a  little  more  than  three 
times  the  length  of  its  diameter;  that  is,  if  a  diameter  is  2 
inches,  the  circumference  will  be  a  little  more  than  6  inches 
long. 


1 68  OBSERVATIONAL   GEOMETRY 

You  can  test  this  by  bending  a  strip  of  paper  around  the  curved  surface  of  a 
cylinder,  noting  the  length,  and  comparing  it  with  the  length  of  the  diameter  of 
the  base. 

Make  the  following  calculations,  supposing  the  length  of  a 
circumference  to  be  three  times  the  length  of  its  diameter: 

1.  Diameter  =  2  cm. ;   circumference  =  ? 

2.  =3   "  "  =? 

3.  "        =  4  "  "  =? 

4.  «         -2  inches ;          "  =  ? 

5.  Radius      =  I  cm. ;  "  =  ? 

6.  •"  =  2   "  "  =  ? 

7.  Circumference  =    6  cm.;  diameter  =  ?;  radius  =  ? 

8.  "  =9   "  "         =  ?        "      =  ? 

9.  "  =3  inches ;  "         =  ?         «      =  ? 

10.  "  =  12        "  "  =  ?  "        =  ? 

Geometers  have  proved  that  the  exact  comparison  between 
a  circumference  and  its  diameter  cannot  be  expressed  in 
numbers ;  and  they  have  accordingly  agreed  to  denote  it  by 
the  Greek  letter  TT  (pronounced  pie).  This  is  expressed  by 
saying  that  a  circumference  has  TT  times  the  length  of  its 
diameter.  TT  is  nearly  equal  to  3^;  that  is,  if  a  diameter  is 
5  cm.  long,  the  circumference  will  be  5  TT  or  about  \$\  cm. 
long. 

Make  the  following  calculations,  considering  TT  to  be  equal 
to  3f:  — 

11.  Diameter  =    i  cm. ;  circumference  =  ? 

12.  "         =    2   "  "  =? 

13.  "         =    3   "  "  =? 

14.  "         =     7  "  "  =? 

15.  Radius      =    I  inch;  "  =? 

16.  "  =    2  inches;        "  =? 

17.  "  =    3      "  "  =? 

18.  Circumference  =  22  cm. ;  diameter  =  ? 

19.  "  =  44   "  radius  =  ? 

20.  "  =  ii  inches;       «     =? 

3.  To  find  the  Length  of  an  Arc.  To  calculate  the  length  of 
an  arc,  you  must  know  the  size  of  the  arc  in  degrees  and  the 
length  of  the  circumference  of  which  the  arc  is  a  part. 


REGULAR  POLYGONS 


169 


Suppose  the  arc  AB  to  be  70°,  and  the  diameter  of  tht 
circle  to  be  3  cm. 

ist.  The  whole  circumference  is  3  TT,  or  3  X  3-}-,  or  9^  cm. 
long. 

2d.  As  the  arc  is  70°  and  the  whole  circumference  con- 
tains 360°,  the  arc  is  g^  or  3^-  of  the  circumference. 

.-.  The  length  of  the  arc  is  ^  X  9f,  or  fa  X  6/>  or  V»  or 
I  £  cm.  long. 

Calculate  the  lengths  of  the  following  arcs,  considering  TT 
to  be  equal  to  3}. 

1.  Arc     35°,  diameter  of  circle    i  cm.       4.    Arc   140°,  radius  of  circle  35  mm. 

2.  Arc     60°,         "  "        7  cm.       5.    Arc     90°,      "  "          4  cm. 

3.  Arc     70°,        "  "      14  cm. 


CHAPTER    XXV 

CONSTRUCTIONS 

I.    To  construct  a  straight  line  which  shall  be  equal  to  a 
given  straight  line. 

(a)    With  the  aid  of  a  graduated  ruler :  — 
A— B 


Let  AB  be  the  given  straight  line : 

Measure  the  length  of  AB  with  a  graduated  ruler,  and  then  draw  XY 
of  the  same  length.     X Y  will  be  the  required  line. 

(b)    With    the    aid    of   compasses    and    an    ungraduated 
ruler: — 

A. -B 


X—  - 

R 

Let  AB  be  the  given  straight  line. 

Draw.^YK  a  straight  line  of  any  convenient  length  evidently  greater 
than  AB. 

With  X  as  a  centre,  and  a  radius  equal  to  AB,  draw  an  arc  /V?  crossing 


XZ  will  be  the  required  straight  line. 

Construct  straight  lines  equal  to  the  following,  with  the  aid 
of  compasses  and  ungraduated  ruler:  — 
-  6  - 


2  -  7 

3  -  8 

4-  --  9 

5  -  10 


COA7S  TR  UCTIONS 


171 


2.    To  bisect  a  Given  Straight  Line.     To   bisect  means  to 
divide  into  two  equal  parts. 

(of)    With  the  aid  of  a  measuring  ruler. 

M 


Let  AB  be  the  given  line. 

Applying  a  graduated  ruler  to  AB,  you  will  find  the  length  to  be  6  cm. 
Divide  this  length  by  2,  and  measure  off  the  quotient  3  cm.  from  A  or 
from  B  to  the  point  M,  which  will  be  the  middle  point  of  AB. 

(£)    With  the  aid  of  compasses  and  ungraduated  ruler. 


V 

A 


M 


Let  AB  be  the  given  line. 

With  A  and  B  as  centres,  and  any  convenient  radius  which  is  evidently 
greater  than  one-half  AB,  draw  arcs  crossing  one  another  at  Cand  D  on 
opposite  sides  of  AB. 

Connect  Cand  D  with  a  straight  line  crossing  AB  at  J/,  which  will 
be  the  middle  point  of  AB. 

Construct  straight  lines  equal  to  the  following,  and  bisect 
them  with  the  aid  of  compasses  and  ungraduated  ruler. 


172 


OBSEK  VA  TIONA  L    GE OME  TR  Y 


3.    To  construct  a  perpendicular  from  a  given  point  to  a 
given  straight  line. 

(a)  With  the  aid  of  a  square:  — 

Let  P  be  the  given  point,  and  AB  the  given 
straight  line. 

Apply   a   square  so    that   one    edge  of   the 
right  angle  may  be  close  to  AB,  and  the  other  X 

edge  close  to  P ;  along  the  latter  edge  draw  a  line  PX  to  AB. 
will  be  the  required  perpendicular. 

(b)  With  the  aid  of  compasses  and  ungraduated  ruler. 


PX 


Let  P  be  the  given  point,  and  AB  the  given  straight  line. 

With  P  as  a  centre,  and  any  convenient  radius  evidently  longer  than 
the  perpendicular  distance  from  P  to  AB,  draw  an  arc  cutting  AB  at  C 
and  D. 

With  C  and  D  as  centres  and  any  convenient  radius  evidently  longer 
than  one-half  CD,  draw  arcs  crossing  one  another  at  E* 

Draw  the  straight  line  PE  crossing  AB  at  X. 

PX  will  be  the  required  perpendicular. 

4.    To  construct  a  perpendicular  to  a  given  straight  line 
from  a  given  point  in  the  line. 

(a)    With  the  aid  of  a  square  :  — 

Let  AB  be  the  given  straight  line  and  P  X 

the  given  point  in  AB. 

Apply  a  square  so  that  the  vertex  of  the 
right  angle  may  be  at  /*,  and  one  edge  may 
lie  close  to  AB.  Along  the  other  edge 


draw  PX,   which  will   be   the   required   per- 
pendicular. 


CONSTX  ACTIONS 
(b~)    With  the  aid  of  a  protractor  and  ruler. 


'73 


Let  AB  be  the  given  straight  line,  and  P  the  given  point  in  AB. 

Apply  the  straight  edge  of  the  protractor  to  AB  so  that  the  notch  may 
be  at  P.-  Then  draw  PX  so  as  to  make  the  angle  BPX  equal  to  90°. 
XP  will  be  the  required  perpendicular. 

(c)    With  the  aid  of  compasses  and  ungraduated  ruler. 


/ 


Let  AB  be  the  given  straight  line,  and  P  the  given  point  in  AB. 

With  P  as  a  centre  and  any  convenient  radius,  draw  an  arc  cutting  AB 
at  C  and  D.  With  C  and  D  as  centres,  and  any  convenient  radius  longer 
than  C/5,  draw  two  arcs  cutting  one  another  at  X.  Draw  the  straight 
line  XP,  which  will  be  the  required  perpendicular. 

5.   To  construct  an  arc  which  shall  be  equal  to  a  given 
arc  both  in  degrees  and  in  length. 
(a)    With  the  aid  of  a  protractor :  — 

This  method  is  explained  on  pp.  42  and  159. 


174  OBSERVATIONAL   GEOMETRY 

(V)    With  the  aid  of  compasses  and  ungraduated  ruler:  — 
B  Z-. w 


Let  A B  be  the  given  arc. 

If  the  centre  O  is  not  given  with  the  arc,  find  it  by  the  aid  of  problem 
9  on  p.  158.  Draw  the  chord  AB  and  the  radius  OB. 

With  any  point  X  as  a  centre,  and  a  radius  equal  to  OB,  draw  an  arc 
YZ  evidently  greater  than  the  required  arc. 

With  Fas  a  centre,  and  a  radius  equal  to  the  chord  AB,  draw  an  arc 
cutting  YZ  at  IV.  YW  will  be  the  required  arc. 

6.   To  construct  an  angle  which  shall  be  equal  to  a  given 
angle. 

(a)    With  the  aid  of  a  protractor:  — 

This  problem  is  explained  on  p.  42. 
(£)    With  the  aid  of  compasses  and  ungraduated  ruler:  — 


Let  ABC\)t  the  given  angle. 

With  B  as  a  centre,  and  any  convenient  radius,  draw  an  arc  DE  be- 
tween the  sides  of  the  angle. 

Then  construct  an  arc  XY  equal  to  DE. 

From  Z,  the  centre  from  which  XY  is  drawn,  draw  the  radii  ZX 
and  ZY. 

XZYvfMl  be  the  required  angle. 


CONSTRUCTIONS 


175 


7.    To  bisect  a  given  arc. 
(a)    With  the  aid  of  a  protractor  and  ruler. 

A 


Let  AB  be  the  given  arc,  and  O  the*  centre  of  its  circle.  Draw  the 
radii  OA  and  OB. 

Measure  the  angle  A  OB  with  a  protractor  and  divide  the  result  by  2. 
Draw  OX  so  as  to  make  an  angle  equal  to  the  quotient,  with  O  as  a 
vertex  and  OA  or  OB  as-  one  side,  and  crossing  AB  at  Y.  Y  will  be  the 
middle  point  of  the  arc  AB. 

(b)  With  the  aid  of  a  square  and 
graduated  ruler. 

Let  AB  be  the  given  arc.  Draw  the  chord 
AB  and  find  its  middle  point  M  with  a 
graduated  ruler.  At  M  draw  XM Y  perpen- 
dicular to  the  chord  AB  and  crossing  the 
arc  AB  at  Z. 

Z  will  be  the  middle  point  of  the  arc  AB. 

(c)    With  the  aid  of  compasses  and  ungraduated  ruler. 

A 


B 


Let  ABbe  the  given  arc. 

Draw  the  chord  AB  and  bisect  it  (as  explained  on  p.  171)  with  the 
line  XY crossing  the  arc  AB  at  Z. 

Z  will  be  the  middle  point  of  the  arc  AB. 


1 76  OBSERVATIONAL    GEOMETRY 

8.    To  bisect  a  given  angle. 
(a)    With  the  aid  of  a  protractor  and  ruler:  — 
The  method  is  similar  to  that  for  bisecting  a  given  arc. 
(£)    With  the  aid  of  a  square  and  graduated  ruler 


B  DA 

Let  ABC  be  the  given  angle. 

Measure  from  B  any  convenient  equal  distances,  BD  on  BA  and  BE 
on  BC. 

Draw  the  straight  line  DE. 

With  a  square  draw  a  perpendicular  ^J/  to  DE. 

BM  will  bisect  the  angle  ABC. 

(^r)    With  the  aid  of  a  graduated  ruler  alone :  — 

Proceed  as  in  case  (b~)  until  the  line  DE  has  been  drawn. 

Then  bisect  DE  with  a  graduated  ruler,  and  connect  the  middle  point 
with  a  straight  line,  which  will  be  the  same  as  the  line  BM  and  will 
bisect  the  angle. 

(d)    With  the  aid  of  compasses  and  ungraduated  ruler:  — 

,C 


B  DA 

Let  ABC-be  the  given  angle. 

With  B  as  a  centre,  and  any  convenient  radius,  draw  the  arc  DE  be- 
tween the  sides  of  the  angle.  Bisect  this  arc  with  the  line  BX>  which 
will  also  bisect  the  angle  ABC. 


CONS  TR  UCTIONS 


177 


9.    To  circumscribe  a  circle  about  a 
square. 

Let  A  BCD  be  a  square. 

Draw  the  diagonals,  and  let  O  be  their  point 
of  intersection. 

With  O  as  a  centre,  and  a  radius  equal  to  OA 
(==  OB—OC  —  OD},  draw  a  circle  which  will 
be  the  required  circle  circumscribed  about  the 
square. 

10.    To  inscribe  a  circle  in  a  square. 

Let  A  BCD  be  a  square. 

Draw  the  diagonals,  and  let  O  be  their  point  of 
intersection. 

With  O  as  a  centre,  and  a  radius  equal  to  one- 
half  a  side  of  the  square,  draw  a  circle  which  will 
be  the  required  circle  inscribed  in  the  square. 

Construct  the  following  squares,  and  draw  circles  in  and 
about  each  :  — 


6. 

7. 

8. 

9. 

10. 


Side  i  inch. 
"      2  inches. 

"     3     " 


II.    To  circumscribe  a  circle  about 
a  triangle. 

Let  ABC  be  a  triangle  of  any  kind. 

Erect  perpendiculars  at  the  middle  points  of 
any  two  sides,  and  prolong  them  until  they 
meet  at  O,  which  will  be  equally  distant  from 
all  three  vertices  of  the  triangle. 

With  O  as  a  centre,  and  a  radius  equal  to  OA 
(=  OB  =  OC),  draw  a  circle,  which  will  be  the 
required  circle  circumscribed  about  the  triangle. 

12.    To  inscribe  a  circle  in  a  triangle. 

Let  ABC  be  a  triangle  of  any  kind. 

Bisect  any  two  of  the  angles,  and  prolong  the  bisectors  until  they  meet 
at  O,  which  will  be  equally  distant  from  all  three  sides. 


78 


OBSERVATIONAL  GEOMETRY 
A 


With  O  as  a  centre,  and  a  radius  equal  to  the  perpendicular  OX 
(=  OY=.  OZ),  draw  a  circle,  which  will  be  the  required  circle  inscribed 
in  the  triangle. 

In  the  case  of  an  equilateral  triangle  the  centre  of  the  in- 
scribed circle  is  the  same  point  as  the  centre  of  the  circum- 
scribed circle. 

Construct  equilateral  triangles  with  the  following  sides,  and 
circumscribe  and  inscribe  a  circle  in  each  case:  — 


1.  Side 

2.  " 

3.  « 

4.  « 

5.  " 


3  cm- 

4  " 

5  " 
25  mm. 

35    " 


6. 
7. 
8. 
9. 
10. 


Side  2  inches. 

"     3      " 

«      4      « 


13.    Miscellaneous  problems  on  constructions. 

A,  B,  C,  and  D  are  the  vertices  of  a  square  circumscribed  about  a  circle. 
With  each  vertex  as  a  centre,  and  a  radius  equal  to  the  radius  of  the  circle, 
an  arc  is  drawn  within  the  circle  and  bounded  by  its  circumference.  Lei 
the  radius  of  the  circle  be  one  inch. 


CONS  TR  UCT10NS 


179 


2.  Construct  a  square.     Then  with  each  vertex  as  a  centre,  and  a  radius  equal 

to  one  of  the  sides,  draw  an  arc  within  the  square  and  bounded  by  its 
sides. 

3.  Construct  a  square  and  draw  its  diagonals.     Then  at  the  middle  points 

of  the  sides,  with  a  radius  equal  to  one-fourth  of  a  diagonal,  draw  circles. 

4.  Construct  a  square.     Then  with  each  vertex  as  a  centre,  and  a  radius  equa 

to  one-half  a  diagonal,  draw  an  arc  within  the  square  and  bounded  by  its 
sides.  Connect  the  ends  of  these  arcs  so  as  to  form  a  regular  octagon. 

5.  A,  />',  C,  D  are  the  vertices  of  a  square,  and  X,  Y,  Z,  W  are  the  middle 

points  of  its  sides.  With  each  of  these  points  as  a  centre,  and  a  radius 
equal  to  one-half  a  side  of  the  square,  arcs  are  drawn  so  as  to  form  the 
annexed  figure.  Let  the  side  of  the  square  be  two  inches. 


6.  Construct  a  square.     Then  with  the  vertices  and  middle  points  of  the  sides 

as  centres,  and  a  radius  equal  to  one-half  a  side,  draw  arcs  outside  the 
square  and  bounded  by  its  sides. 

7.  Construct  a  square.     Then   upon  each  side   as  a  diameter  draw  a  semi- 

circumference  within  the  square. 

8.  Construct  a  square  and  draw  its  diagonals.     With  the  vertices  as  centres, 

and  a  radius  equal  to  one-fourth  of  a  diagonal,  draw  arcs  within  the  square 
and  bounded  by  its  sides.  With  the  point  of  intersection  of  the  diagonals 
as  a  centre,  and  the  same  radius  as  before,  draw  a  circumference,  which 
will  be  tangent  to  the  other  arcs. 

9.  Construct  a  circle  and  determine  the  vertices  of  an  equilateral   inscribed 

triangle.  With  each  of  these  points  as  a  centre,  and  a  radius  equal  to  the 
radius  of  the  circle,  draw  an  arc  within  the  circle  and  bounded  by  the 
circumference. 

10.  Construct  an  equilateral  triangle.    Then  with  each  vertex  as  a  centre,  and 

a  radius  equal  to  a  side  of  the  triangle,  draw  an  arc  between  the  other  two 
vertices. 

11.  A,  B,  and  C  are  the  vertices  of  an  inscribed  equilateral  triangle,  and  OA, 

OB,  and  OC  are  radii.     Upon  these  radii  as  diameters  arcs  are  drawn  so 


I8o  OBSERVATIONAL   GEOMETRY 

as  to  meet  one  another  as  in  the  annexed  figure.     Let  the  radius  of  the 
circle  be  two  centimetres,  and  construct  the  figure. 


12.  Construct  an  equilateral  triangle.     Then  with  each  vertex  as  a  centre,  and 

a  radius  equal  to  one-half  a  side  of  the  triangle,  describe  a  circumference. 
The  three  circles  will  be  tangent  to  one  another. 

13.  Construct  a  circle  and  determine  the  vertices  of  a  regular  inscribed  hexagon. 

With  these  points  as  centres,  and  a  radius  equal  to  the  radius  of  the  circle, 
draw  arcs  within  the  circle  and  bounded  by  the  circumference. 

14.  ABCDEF  is  a  regular  hexagon,  whose  diagonals  cross  each  other  at  O. 

With  O  and  each  vertex  of  the  hexagon  as  centres,  and  a  radius  equal  to 
one-half  a  side  of  the  hexagon,  circles  are  constructed,  six  of  which  are 
tangent  to  the  seventh.  Let  the  side  of  the  hexagon  be  <?ne  inch,  and 
construct  the  figure. 


15.  Construct  a  circle  and  inscribe  in  it  a  regular  hexagon.     Upon  the  sides  of 

the  hexagon  as  diameters  construct  circles. 

16.  A,  B,  C,  and  D  are  the  vertices  of  a  square.     Upon  the  sides  of  the  square 

as  diameters  semicircumferences  are  drawn  inwardly.     With  the  vertices 
of  the  square  as  centres,  and  a  radius  equal  to  a  side,  arcs  are  drawn  out- 


CONSTRUCTIONS 


181 


wardly  until  they  meet.  Let  the  side  of  the  square  be  3  cm.,  and  construct 
the  figure. 

17.  Determine  the  vertices  of  a  square,  and  construct  a  figure  having  the  arcs 

of  the  preceding  problem  reversed  in  position,  the  semicircumferences 
being  drawn  outwardly  and  the  other  arcs  inwardly. 

18.  Construct  a  circle  and  determine  the  vertices  of  a  regular  inscribed  dodeca- 

gon. Omitting  every  third  vertex,  with  the  rest  of  the  vertices  as  centres, 
and  a  radius  equal  to  the  radius  of  the  circle,  draw  arcs  from  the  circum- 
ference to  the  centre. 

19.  Construct  an  equilateral  triangle.     At  the  middle  points  of  the  sides  draw 

perpendiculars  to  meet  at  a  point,  and  prolong  each  in  the  opposite  direc- 
tion so  that  the  part  outside  may  be  equal  to  the  part  within  the  triangle. 
With  the  outer  ends  of  these  perpendiculars  as  centres,  draw  arcs  outside 
the  triangle  with  the  sides  for  chords. 

20.  AD  is  the  diameter  of  a  circle  and  is  divided  into  three  equal  parts  at  B 

and  C.  Upon  AB,  AC,  BDt  and  CD  as  diameters  semicircumferences  are 
drawn,  two  on  each  side  of  the  diameter.  Construct  the  figure,  taking  6 
cm.  as  the  diameter  of  the  circle. 


1 82 


OBSER  VA  TIONA  L    GEOME  TR  Y 


21.  Construct  a  circle  with  a  diameter  of  8  cm.  divided  into  four  equal  parts, 

and  draw  four  semicircumferences  so  as  to  form  a  figure  like  that  of  the 
preceding  problem. 

22.  Construct  a  circle  and  determine  the  vertices  of  an  inscribed  square.     With 

these  points  as  centres,  and  a  radius  equal  to  the  radius  of  the  circle,  draw 
semicircumferences  which  will  all  pass  through  the  centre  of  the  circle  and 
meet  so  as  to  form  a  four-leaved  figure. 

23.  Determine  the  vertices  of  an  equilateral  triangle.     Then  with  these  points 

as  centres,  and  a  radius  equal  to  one-half  a  side  of  the  triangle,  draw 
circles. 

24.  Construct  a  figure  similar  to  the  first  below.     A,  B,  and  C  are  the  vertices 

of  an  equilateral  triangle  ;  each  side  should  be  4  cm.  Xy  Y}  and  Z  are  the 
middle  points  of  the  sides. 


25.  Construct  a  figure  similar  to  the  second  above.     A,  B,  and  C  are  the  vertices 

of  an  equilateral  triangle  ;  X,  Yt  and  Z  are  ihe  middle  points  of  the  sides. 
Three  semicircumferences  are  drawn  upon  the  sides,  and  six  arcs  having 
the  vertices  for  centres  and  radii  equal  to  one  of  the  sides.  Let  the  dis- 
tance from  A  to  B  be  \\  inches. 

26.  Construct  a  figure  having  the  same  arcs  as  in  problem  25,  but  reverse  their 

positions,  so  that  the  semicircumferences  will  be  drawn  outwardly  and  the 
other  arcs  inwardly. 

27.  Construct  a  circle  and  determine  the  vertices  of  a  regular  inscribed  octagon. 

With  these  points  as  centres,  and  a  radius  equal  to  a  side  of  the  octagon, 
draw  arcs  within  the  circle  and  bounded  by  the  circumference. 


CHAPTER    XXVI 

AREAS 

Areas  of  Polygons.     Review  what  is  said  about  areas  on  pp. 
13,  H- 

FOR   REFERENCE 

For  measurement  of  areas,  two  tables  are  in  common  use,  the  Metric 
and  the  English. 

METRIC   TABLE 

100  sq.  millimetres  (sq.  mm.)  =  I  sq.  centimetre  (sq.  cm.)  =J  sq.  inch, 

nearly. 

100  sq.  centimetres  =  I  sq.  decimetre  (sq.  dcm.)  =  ^  sq  ft.  nearly. 
100  sq.  decimetres  =  I  sq.  metre  (sq.  m.)  =  I  centar  (ca.)  =  i£  sq.  yds. 

nearly. 
100  sq.  metres  =  I  sq.  dekametre  (sq.  dkm.)  =  i   ar  (a)  =  4  sq.   rods 

nearly. 
100  sq.  dekametres  =  i  sq.  hektometre  (sq.  hkm.)  =  I  hektar  (hka)  = 

2^  acres  nearly. 
100  sq.  hektometres  =  i  sq.  kilometre  (sq.  km.)  =  |  sq.  mile  nearly. 

ENGLISH   TABLE 

144  sq.  inches  (sq.  in.)  =  i  sq.   foot    (sq.  ft.)  =  9^ sq.  dcm.  nearly. 

9  sq.  feet  =  i  sq.  yard  (sq.  yd.)  =  |  sq.  m.          " 

30^-  sq.  yards  =  272^  sq.  ft.  =  I  sq.  rod    (sq.   rd.)  =  25^  sq.  m.      " 
1 60  sq.  rods  =  i  acre  =  40^  ars  " 

640  acres  =  i  sq.  mile  =  2|  sq.  km.      " 

I.  Area  of  a  Rectangle.     Review  the  explanation  of  the 
area  of  rectangles   on   p.  28. 

1.  A  piece  of  paper  is  7  cm.  long  and  4  cm.  wide,  having  the  shape  of  a  rect- 
angle. Draw  a  plan  of  the  exact  size ;  divide  it  into  square  centimetres 
by  drawing  lines ;  then  count  the  squares,  writing  inside  each  square  its 
number  from  one  upwards. 


184  OBSERVATIONAL    GEOMETRY 

2.  Draw  a  rectangle  8  cm.  long  and  2  cm.  5  mm.  wide,  and  draw  lines  dividing 

it  into  squares  and  parts  of  squares.  How  many  sq.  centimetres  does  it 
contain  ?  Then  taking  scissors  cut  up  the  rectangle  into  the  parts  which 
you  marked  off,  match  together  the  parts  of  squares,  and  see  how  many 
squares  there  are  altogether.  Is  the  number  the  same  as  you  found  before 
by  calculation  ? 

3.  Do  the  same  as  in  the  preceding  question  with  a  rectangle  8  inches  long 

and  i \  inches  wide.  How  many  of  the  parts  of  squares  must  you  put 
together  in  order  to  make  a  complete  square  ? 

4.  Do  the  same  as  in  the  preceding  question  with  a  rectangle  3^  inches  long 

and  2\  inches  wide.  In  this  case  one  of  the  squares  will  be  incomplete  : 
how  much  does  it  lack  to  become  a  complete  square  ? 

5.  A  "  shuffle  board  "  is  30  feet  long  and  20  inches  wide,  and  has  the  shape  of 

a  rectangle.     What  would  you  consider  to  be  the  most  convenient  unit  in 

which  to  calculate  its  area? 

Draw  a  plan  of  the  shuffle  board  on  the  scale  of  ^ ;  that  is,  make  each  edge 
of  your  rectangle  one-fortieth  of  the  corresponding  edge  of  the  shuffle  board. 
Then  calculate  the  area  of  the  board  without  drawing  lines  to  divide  the  figure 
into  units.  Give  the  answer  both  in  square  feet  and  in  square  inches  :  does  the 
result  change  or  confirm  your  opinion  as  to  the  best  unit  to  use  in  this  case  ? 

6.  A  "  bagatelle  board  "  is  i  metre  5  clem,  long  and  7  decimetres  wide.     Draw 

a  plan  of  the  board  on  the  scale  of  j^,  and  calculate  the  area  of  the  board 
both  in  square  metres  and  in  square  decimetres.  How  does  the  area  of 
the  board  compare  with  the  area  of  your  plan  ? 

7.  A  cricket  field  is  70  yards  long  and  50  yards  wide,  and  has  the  shape  of  a 

rectangle.     Draw  a  plan  of  the  field  on  the  scale  of  20  yards  to  the  inch; 

that  is,  represent  a  length  of  20  yards  on  your  plan  by  one  inch. 
What  is  the  area  of  your  plan  ? 
What  is  the  area  of  the  field  ? 

8.  A  surveyor  finds  that  a  piece  of  land  extends  northeast  150  metres,  then 

northwest  60  metres,  then  southwest  150  metres,  and  lastly  southeast  60 
metres.  Draw  a  plan  of  the  land  on  the  scale  of  yoVi5>  anc^  ^n<^  tne  areas 
of  the  plan  and  the  land. 

9.  A  tennis  court  for  the  single-handed  game  is  a  rectangle  27  feet  wide  and 

78  feet  long.     The  net  crosses  the  middle  points  of  the  side  lines ;  that  is, 
the  two  longer  sides.     The  half-court  line  connects  the  middle  points  of 
the  base-lines  ;  that  is,  the  two  shorter  sides.     The  two  service  lines  are 
drawn  parallel  to  the  base-lines,  and  each  is  21  feet  from  the  net. 
Draw  a  diagram  of  the  court  on  the  scale  of  ^^ ;   that  is,  you  will  represent 
a  length  of  eighteen  feet  by  one  inch  on  your  plan.     Then  calculate :  — 

(a)  The  area  of  your  plan. 

(b)  The  areas  of  the  eight  divisions  of  your  plan. 

(c)  The  area  of  the  court. 

(d)  The  areas  of  the  eight  divisions  of  the  court. 

10.  A  tennis  court  for  the  four-handed  game  has  the  same  length  as  the  court 
for  the  single-handed  game,  but  is  36  feet  wide  instead  of  27  feet.  How 
much  additional  area  does  it  contain  ? 


AREAS 


185 


2.   Area  of  a  Parallelogram.     The  area  of  a  parallelogram 
is  equal  to  the  product  of  its  base  and  altitude. 


The  base  is  any  one  of  the  sides. 

The  altitude  is  the  perpendicular  distance  between  the  base 
and  the  opposite  side. 

Thus  in  ABCD,  AB  is  the  base,  and  DP  the  altitude. 

Draw  accurately  on  paper  a  parallelogram  ABCD  with  any  convenient 
sides  and  angles.     Take  AB  as  the  base,  and  draw  the  altitude  DP. 


B 


Cut  the  parallelogram  out  from  the  paper.  Then  cut  off  the  triangle 
ADP,  and  match  it  upon  the  figure  in  the  position  BCP( '. 

You  can  paste  the  two  parts  together  with  a  strip  of  paper  on  the  back. 

You  have  now  changed  the  parallelogram  into  a  rectangle,  keeping  the 
same  base  and  altitude. 

As  the  area  of  the  rectangle  is  the  product  of  its  base  and 
altitude,  this  is  also  the  area  of  the  parallelogram. 

Draw  the  following  parallelograms  according  to  the  de- 
scriptions given,  and  calculate  their  areas.  Write  the  values 
of  the  given  parts  and  the  area  in  and  about  the  diagrams. 


i86 


OBSERVATIONAL   GEOMETRY 

3  cm. 


§          Area  -  6  sq.  cm. 


1.  Two  opposite  sides  each  3  cm.  long,  and  2  cm.  apart. 

2.  Two  opposite  sides  each  4  cm.  long,  and  i  cm.  apart. 


3   " 

2     " 


9. 


5    A  base  2  cm.  5  mm.  long,  distant  i  cm.  from  the  opposite  side. 

6.  "     "     4    "  "  "         2  "  4  mm.  from  the  opposite  side. 

7.  "    "      i    "    8    "       "          "        3  «  2    "        "      " 

8.  Two  sides  each  8  cm.  long,  two  sides  each  4  cm.  long,  two  angles  of  45°, 

and  2  angles  of  135°. 

Two  sides  each  6  cm.  long,  two  sides  each  4  cm.  long,  two  angles  of  60°, 
and  two  angles  of  120°. 

10.  Two  sides  of  6  cm.  4  mm.  and  4  cm.,  making  with  each  other  an  angle  of 

150°. 

11.  Sides  of  3  and  8  cm.,  inclined  to  each  other  at  an  angle  of  80°. 

12.  A  surveyor  marks  off  a  line  on  the  ground  running  due  east,  8  metres  long. 

From  the  east  end  he  draws  a  line  5  metres  long,  running  northwest,  thus 
making  an  angle  of  45°  with  the  first  line.  From  the  north  end  of  the 
second  line  he  draws  a  line  due  west,  8  metres  long.  Lastly  he  draws  a 
line  connecting  the  west  ends  of  the  first  and  third  lines. 

Draw  a  plan  of  the  enclosed  land  (scale  ifa),  and  find  :  — 

(«)    The  direction  in  which  the  fourth  line  runs. 

(b)  The  angles  which  the  fourth  line  makes  with  the  first  and  third. 

(c)  The  length  of  the  fourth  line  as  it  appears  in  your  plan. 
(a)   The  real  length  of  the  line  on  the  ground. 

(e)    The  class  to  which  the  figure  belongs. 

(_/")  The  area  of  your  plan. 

(g)    The  real  area  of  the  ground. 

3.    Area  of  a  Triangle.     The  area  of  a  triangle  is  equal  to 
one-half  the  product  of  its  base  and  altitude. 

C 


AREAS 


187 


The  base  is  any  one  of  its  sides. 

The  altitude  is  the  perpendicular  distance  from  the  base  to 
the  vertex  of  the  opposite  angle. 

Thus  in  the  triangle  ABC,  AB  is  the  base,  and  CP  is  the 
altitude. 

Draw  on  paper  a  parallelogram  ABCD  of  any  convenient  sides  and 
angles.  Taking  AB  for  the  base  draw  the  altitude  DP.  Draw  the 
diagonal  DB. 


Cut  out  the  parallelogram  from  the  paper,  and  cut  it  into  two  parts 
along  the  diagonal  DP. 

Then  turn  one  part  around,  place  it  directly  over  the  other,  and  you 
will  see  that  the  two  are  equal,  and  the  triangle  is  one-half  the  parallelo- 
gram. 

Now  the  area  of  the  parallelogram  is  the  product  of  its  base 
and  altitude.  So  the  area  of  the  triangle,  which  is  one-half 
the  parallelogram,  is  one-half  the  product  of  the  base  and 
altitude  of  the  parallelogram;  that  is,  one-half  the  product  of 
its  own  base  and  altitude. 

Construct  the  following  triangles  and  calculate  their  areas, 
writing  in  the  values  upon  the  diagrams :  — 

1.   Base  8  cm.,  altitude  4  cm. 

2  «     4  «          «         8  " 

3  «     5   ,.          «         3  « 

4.  "     3  "         "         5  " 

5.  A  right  isosceles  triangle  whose  equal  sides  are  each  5  cm.  long. 

6.  A  right  triangle,  the  sides  of  the  right  angle  being  5  cm.  and  3  cm. 

7.  An  isosceles  triangle,  whose  equal  angles  are  each  45°,  and  whose  equal 

sides  are  each  5  cm.  long. 


1 88 


OBSERVATIONAL   GEOMETRY 


8.  Draw  a  right  triangle.     Then  draw  two  more  lines:  — 
(<?)    So  as  to  form  a  rectangle. 

(b)    So  as  to  form  a  parallelogram. 

9.  Draw  an  isosceles  triangle.     Then  draw  two  more  lines :  — 
(a)    So'  as  to  form  a  rhombus. 

(b}    So  as  to  form  a  parallelogram. 
10.    Draw  a  right  isosceles  triangle.     Then  draw  two  more  lines :  — 

(a)  So  as  to  form  a  square. 

(b)  So  as  to  form  a  rhombus. 

4.  Area  of  a  Trapezoid.  The  area  of  a  trapezoid  is  equal 
to  its  altitude  multiplied  by  one-half  the  sum  of  the  parallel 
sides. 

The  altitude  of  a  trapezoid  is  the  perpendicular  distance 
between  the  parallel  sides. 

D 


Thus  in  the  trapezoid  A  BCD,  AB  and  CD  are  the  parallel  sides,  and  Z>Pis 
the  altitude.  The  parallel  sides  are  also  called  bases. 

Draw  on  paper  a  trapezoid  A  BCD,  in  which  AB  and  CD  are  the  par- 
allel sides.  Draw  MR  connecting  the  middle  points  of  AD  and  BC. 
Lay  off  on  AB  the  distance  AS  equal  to  MR.  Draw  SR. 

Cut  out  the  trapezoid  from  the  paper;  cutoff  the  triangla  SBR  and 
place  it  in  the  position  S'CR;  paste  the  parts  together. 

You  have  now  changed  the  trapezoid  into  a  parallelogram. 

The  two  parallel  sides  of  the  trapezoid  have  become  equal 
sides  of  the  parallelogram,  and  one-half  the  sum  of  the  parallel 
sides  is  equal  to  AS,  the  base  of  the  parallelogram. 

The  altitude  DP  is  unchanged. 

Now,  since  the  area  of  the  parallelogram  is  the  product  of 
its  altitude  and  base,  the  area  of  the  trapezoid  is  the  product 
of  its  altitude  and  one-half  the  sum  of  its  bases. 


AREAS 


189 


Draw   the  following  trapezoids  and   calculate  their  areas, 
writing  in  the  values  upon  the  diagrams:  — 


1.  Bases  4  cm.  and  2  cm.  ;  altitude  3  cm. 

2.  "       3  cm.  5  mm.  and  2  cm.  7  mm.;  altitude  2  cm.  4  mm. 

3  a          5     «      2      «  «      3     .«       6      «  .,  j     (i 

4.  The  sides  and  angles  of  a  trapezoid  taken  in  order  are  7  cm.,  20°,  i  cm.  8 

mm.,  iCo°,  4  cm.  6  mm.,  140°,  9  mm.,  40°. 

5.  A  face  of  the  pedestal  of  a  statue  has  the  shape  of  a  trapezoid.     The  par- 

allel sides  of  the  trapezoid  are  6  metres  and  4  metres  long  ;  the  other 
sides  are  each  2  metres  long.  The  angles  at  the  ends  of  the  longest  side 
are  each  60°.  Draw  a  plan  of  the  trapezoid  on  the  scale  of  y£ff,  and  cal- 
culate the  area  of  the  real  figure. 

6.  A  plot  of  ground  has  the  shape  of  a  trapezoid.     The  longer  base  is  48 

metres,  and  makes  an  angle  of  30°  with  each  of  the  sides  which  it  meets, 
these  being  each  24  metres  long. 
Draw  a  plan  of  the  ground  on  the  scale  of  T^,  ar»d  calculate  :  — 

(a)  The  length  of  the  fourth  side  on  your  plan. 

(b)  The  area  of  the  ground. 

7.  Some  boys,  who  are  measuring  the  dimensions  of  a  platform  which  has  the 

shape  of  the  frustum  of  a  pyramid,  find  that  the  slant  height  is  I  ft.  8  in. 
on  all  sides.  The  perimeter  of  the  lower  base  is  14  ft.  and  that  of  the 


How  many  square  feet  of  boards  are  there  in  the  platform  ? 


190 


OBSERVATIONAL    GEOMETRY 


upper  base  is  n  ft.     The  bases  are   rectangles,  one  edge  of  the  upper 
base  being  3  ft. 

How  many  square  feet  are  there   in  the  lateral  surface  and  upper  base 
together  ? 

8.    The  bank  of  a  reservoir  has  the  shape  of  a  trapezoid.      The  upper  and 
lower  edges  are  20  and  30  metres  long,  and  5  metres  apart.     The  other  two 
edges  make  each  an  angle  of  45°  with  the  lower  edge. 
Draw  a  diagram,  naming  your  own  scale,  and  calculate  :  — 
(a]    The  lengths  of  the  two  edges  which  are  not  parallel. 
(l>)    The  angles  which  those  edges  make  with  the  upper  edge. 
(<:)    The  area  of  the  reservoir's  bank. 

5.  Area  of  aPolygon.  The  area  of  a  polygon  can  be  estimated 
by  the  aid  of  transparent  paper  ruled  in  little  squares  whose 
area  has  been  measured.  Lay  the  paper  over  the  polygon, 


mm 

Sffffln 

ipff- 


H44^4-HH-H44+ 


H-H-B-T- 

j i        II 

i  ~i     r  "r^T~T 
h-H-H-H-Hi 


i 

- 

-- 

- 


-i-t-t-t-t-|-t-i-i-!-t1-l-i-t--H-i-- 


count  the  number  of  whole  squares  which  are  within  the  pe- 
rimeter, and  estimate  the  sizes  of  the  parts  of  squares.  The 
sum  of  all  is  the  area  of  the  polygon. 

The  paper  in  the  figure  is  ruled  in  squares  with  an  edge  -fe  of  an  inch  long :  — 
What  is  the  area  of  one  of  the  squares  ? 
Can  you  see  that  BC  is  the  diagonal  of  a  certain  square  ? 
Can  you  see  that  A G  is  the  diagonal  of  a  rectangle?     If  so,  you  may  find 
exactly  what  are  the  sums  of  the  partial  squares  which  adjoin  these  lines. 
What  is  the  area  of  the  whole  polygon  ABCDEFG? 


AREAS 


191 


This  method  is  useful  in  finding  roughly  the  areas  of 
countries,  states,  townships,  etc.,  from  maps.  Suppose,  for 
instance,  that  you  wish  to  find  the  area  of  the  State  of  Penn- 
sylvania from  a  map.  First,  you  must  ascertain  the  scale  on 
which  the  map  is  drawn;  this  is  given  by  the  line  XY  as  100 
miles  to  the  inch.  If,  therefore,  your  measuring  paper  is 


44-UH-   i$R-Hi*i    -^bbn** 

ttitiit ^&±j±j±ht^y ^ 


100   MILES  TO  AN    INCH 


ruled  in  squares  with  edges  -y^  of  an  inch  long,  each  square 
represents  an  area  of  100  square  miles.  Then,  applying  your 
measuring  paper,  you  will  notice  that  the  part  of  the  map 
which  is  indicated  by  the  letters  ABCD  is  a  rectangle  enclos- 
ing 25  X  15  or  375  of  the  squares.  The  rest  of  the  area  is 
covered  by  squares  and  parts  of  squares  which  together  make 
about  75  more,  and  added  to  the  others  give  450  for  the  total 
number.  Since  each  square  represents  an  area  of  100  square 
miles,  the  area  of  Pennsylvania  is  about  45 ,000  square  miles. 

There  are  several  methods  of  calculating  the  area  of  a 
polygon  with  greater  accuracy. 

1st.  The  polygon  can  be  divided  into  triangles,  whose 
areas  are  found  separately  and  then  added. 


192 


OBSER  VA  TIONA  L    GEOME  TR  Y 


The  bases  of  the  triangles  are  diagonals  drawn  from  one  of 
the  vertices  of  the  polygon;  the  altitudes  are  perpendiculars 
drawn  to  the  diagonals  from  the  opposite  vertices  of  the 
triangles. 


1.   Calculate  the  area  of  the  polygon  ABCDEF  from  the  following  measure- 
ments :  — 

AC=  ii  rods;  AD  =  16  rods;  AE  =  II  rods;  BX=z  rods;  CY—  4  rods; 
£Z  =  6rods;  FW  =  4  rods. 


2.    Calculate  the  area  of  the  polygon  ABCDEF irom  the  following  measure- 
ments :  — 

AC  =  TO  metres ;  AD  =  17  metres  ;  AE  =  13  metres  ;  BX  =  4  metres  ;  CY 
=  6  metres ;  EZ  =  6  metres ;  FW=  5  metres. 


AREAS 


'93 


3.   Calculate  the  area  of  the  polygon  ABCDEFG  from  the  following  measure- 
ments :  — 

AC  =  10  metres;  AD  =  n  metres;  AE  =  9  metres;  AF=  8  metres;  BX= 
5  metres;  CY=  3  metres;  EZ—  5  metres;  FW '=  5  metres;  GV—2  metres. 

2d.  The  area  of  a  polygon  can  be  found  by  drawing  its 
longest  diagonal  and  perpendiculars  upon  this  diagonal  from 
the  vertices.  The  polygon  is  thus  divided  into  trapezoids, 
rectangles,  or  right  triangles,  whose  areas  are  found  separately 
and  then  added. 


Calculate  the  area  of  the  polygon  ABCDEFG  from  the  following  measure- 
ments :  — 

BX=  6  metres;  CZ  =  7  metres;  DV=$  metres;  GY—  5  metres;  FW= 
5  metres. 

AX  —  4  metres  \XY~  2  metres;  YZ —  I  metre;  ZW=  3  metres;  WV— 
2  metres  ;  VE  —  2  metres. 


194 


OBSERVATIONAL   GEOMETRY 


3d.  The  area  of  a  polygon  can  be  found  by  a  method 
commonly  used  by  land  surveyors. 

A  line,  LNt  called  "  the  base  line,"  is  drawn  at  one  of  the 
vertices,  and  perpendiculars  are  drawn  to  this  from  the  other 
vertices,  forming  trapezoids,  rectangles,  or  right  triangles, 


w 


whose  areas  are  found  separately  and  added.  Then  from  the 
sum  there  is  subtracted  the  areas  of  the  parts  which  lie  out- 
side the  polygon.  In  this  figure  the  base  line  LN  is  drawn 
perpendicular  to  the  diagonal  AE.  The  parts  to  be  sub- 
tracted from  the  whole  area  consist  of  a  trapezoid  and  two 
right  triangles. 

Calculate  the  area  of  the  polygon  ABCDEFG  from  the  following  measure- 
ments :  — 

CX—  7  metres  ;  BY—  4  metres  ;  DZ  —  12  metres  ;  EA  =  14  metres;  FW 
=  10  metres;  GW—  6  metres  ;  XY=i  metre;  YZ=  2  metres;  ZA  —  4  metres; 
A  W—  5  metres. 

Compare  the  result  with  that  of  the  preceding  problem,  as  the  two  polygons 
are  the  same. 


AREAS  195 


6.  Area  of  a  Circle.  The  area  of  a  circle  can  be  found  by 
calculating  the  length  of  the  circumference,  multiplying  that 
by  the  length  of  the  radius,  and  dividing  the  product  by  2. 


This  rule  depends  upon  regarding  the  area  of  the  circle  as 
equal  to  the  sum  of  the  areas  of  a  number  of  equal  isosceles 
triangles  whose  bases  are  chords  and  whose  vertices  opposite 
to  the  chords  meet  at  the  centre  of  the  circle. 

If  there  were  only  six  of  these  triangles,  composing  a  hexagon,  as  in  figure  I, 
there  would  be  considerable  difference  between  the  area  of  the  circle  and  that 
of  the  polygon.  But  if  the  number  of  triangles  were  increased  only  to  twenty- 
four,  as  in  figure  2,  the  area  of  the  polygon  would  approach  much  nearer  the 
area  of  the  circle.  It  is  to  be  noticed  also  that  the  altitudes  of  the  triangles  in 
figure  2  are  each  nearly  equal  to  the  radius  of  the  circle  ;  and  the  sum  of  the 
bases  is  nearly  equal  to  the  circumference  of  the  circle.  If  the  number  of 
triangles  were  further  increased,  they  would  form  a  polygon  which  could  hardly 
be  distinguished  from  the  circle,  though  there  would  always  be  a  difference. 

Now  the  sum  of  the  areas  of  the  triangles  can  be  found  by 
multiplying  the  sum  of  their  bases  by  their  altitude,  and 
dividing  the  product  by  2. 

So  the  area  of  the  circle  can  be  found  by  multiplying  its 
circumference  by  its  radius,  and  dividing  the  product  by  2. 

Suppose  the  radius  of  a  circle  to  be  4  cm. 

Then  the  circumference  =  2  X  3^  X  4  —  25^  cm. 

And  the  area  =  4  X  25^-^2  =  5  of  sq.  cm. 

Calculate  the  areas  of  the  following  circles,  considering  IT  to  be  equal  to  3f :  — 

1.  Radius  =    7  cm.  6.    Diameter  =  10  cm. 

2.  =    3    "  7.  =     2  cm. 

3.  "       =  14  «  8.  =  12  " 

4.  =    i  inch.  9.  "        =    4  inches. 

5.  "       =2  inches.  10.  "        =8  inches. 


196  OBSERVATIONAL   GEOMETRY 

7.    Sector.     A  sector  is  a  part  of  a  circle  included  by  two 
radii  and  an  arc,  as  A  OB. 

A  sector  is  often  described  by  the  size  of  the  angle  between 
the  radii ;  thus  if  the  angle  AOB  is  45°,  the  sector  is  called 
"a  sector  of  45°." 


A  sector  of  any  required  size  is  constructed  by  drawing 
two  radii  forming  an  angle  of  the  size  indicated. 
Construct  the  following  sectors. 

1      45°,  radius  2  cm.  6.      30°,  radius    i  inch. 

2.  120°,      "      3  "  7.     60°,     "         2  inches. 

3.  90°,      "      4  "  8-     45°.      "       i*      " 

4.  100°,      "      2   "  9.     90°,      "        2      " 

5.  20°,      "      4  "  10.    120°,      "       i£      " 

To  find  the  area  of  a  sector :  — 

(a)  When  the  length  of  the  radius  and  the  length  of  the 
arc  are  known. 

Like  the  whole  circle,  the  sector  can  be  regarded  as  composed  of  a 
countless  number  of  triangles  whose  altitude  is  the  radius  and  the  sum 
of  whose  bases  is  the  arc. 

The  area  of  the  sector,  therefore,  is  found  by  multiplying  the  length  of 
the  arc  by  the  length  of  the  radius,  and  dividing  the  product  by  2. 

Thus,  if  the  radius  is  ij  cm.  and  the  arc  2  cm.,  the  area  of  the  sector 
will  bs  |  X  2  -r-  2  —  |  sq.  cm. 

(b)  When  the  length  of  the  radius  and  the  angle  of  the 
sector   are   known. 

Let  the  radius  be  i^  cm.,  and  the  angle  50°. 

The  sector  is  ^5-  or  ^  of  the  whole  circle. 

The  area  of  the  circle  is  ^  ?r  or  7^-. 

.-.    The  area  of  the  sector  is  fo  X  7^  =  -fa  X  f |  =  ff  sq.  cm. 


AKEAS  197 

(c)  When  the  length  of  the  radius  and  the  number  of 
degrees  in  the  arc  are  known. 

Since  the  arc  and  the  angle  formed  by  the  radii  have  the  same  number 
of  degrees,  the  method  of  finding  the  area  is  the  same  as  in  case  (£). 

Calculate  the  areas  of  the  following  sectors :  — 

11.  Radius  4  cm.,  arc  3  cm.  16.    Radius  2  cm.,  angle      30°. 

12.  "  5  •'      "8   "  17.  "  4  "  "  45°- 

13.  "  2  "       "i    "  18.  "  7    "  arc  90°. 
14  "  3  "       "5   "  19.  "  3  "  "  100°. 
15.  "  7  "angle  60°.  20.  "  i    "  "  120°. 

8.    Segment.     A  segment  of  a  circle  is  a  part  of  the  circle 
enclosed  between  a  chord  and  its  arc. 

The  word  segment  means  "  a  part  cut  off." 

The  size  is  often  designated  by  the  number  of  degrees  in 
its  arc ;  thus  if  the  arc  is  60°,  the  segment  is  called  "  a  seg- 
ment of  6o°." 


The  area  of  a  segment  can  be  found  by  drawing  radii  to 
the  ends  of  the  arc  and  deducting  the  area  of  the  triangle 
from  the  area  of  the  sector  thus  formed. 

Construct  the  following  segments :  — 

1.  Radius    2  cm.,       arc    80°.  4.  Radius      i  inch,   arc     90°. 

2.  "         3   "  "      90°.  5.        "        i£  inches  "      75°. 

3.  "        25  mm.      "     120°. 

Calculate  the  areas  of  the  following  segments :  — 

6.  Radius       2     cm.,        arc     90°.  9.    Radius        i  inch,          arc    90°. 

7.  "  3      "  "      90°.  10.    Diameter   4  inches,  90°. 

8.  Diameter  4      "  "      90°. 


I9S  OBSERVATIONAL   GEOMETRY 

9.  Surface  of  a  Sphere.  The  surface  of  a  sphere  is  exactly 
equivalent  to  the  areas  of  four  circles  of  the  same  diameter 
as  the  sphere  (see  p.  113). 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  7  cm.  ? 

2.  What  is  the  surface  of  a  sphere  whose  radius  is  5  cm.? 

3.  The  diameter  of  the  moon  is  about  2160  miles.     How  many  square  miles 

are  there  on  its  surface  ? 

4.  How  much  would  it  cost  at  2  cents  a  square  foot  to  paint  a  hemispherical 

dome  whose  diameter  is  44  ft.  ? 

5.  What  is  the  surface  of  the  greatest  globe  which  can  be  cut  from  a  cubical 

block  of  wood  whose  edge  is  I  dcm.  ? 

6.  What  is  the  diameter  of  a  sphere  whose  circumference  is  22  cm.? 

7.  How  many  square  inches  of  leather  will  it  take  to  cover  a  ball  whose  cir- 

cumference is  9  inches  ? 

8.  How  many  balls  each  5  cm.  in  diameter  can  be  covered  from  a  square  metre 

of  cloth  ? 

9.  How  does  the  surface  of  a  sphere  compare  with  that  of  a  cube  whose  edge 

is  equal  to  a  diameter  of  the  sphere  ? 

10.    How  does  the  surface  of  a  sphere  compare  with  the  lateral  surface  of  a 
cylinder  which  will  exactly  contain  the  sphere? 


CHAPTER    XXVII 

VOLUMES 
Volume.     Review  what  is  said  about  volume  on  p.  13. 

(FOR   REFERENCE) 

For  measurement  of  volumes  two  tables  are  in  common 
use,  the  Metric  and  the  English. 

METRIC   TABLE 

1000  cubic  millimetres  (cu.  mm.)  =  i  cubic  centimetre  (cu.  cm.)  =  /0  cu.in.  nearly, 
looo  cu.  centimetres  —  i  cu.  decimetre  (cu.  dcm.)  =  i  litre  =^Gca.  ft.      " 
1000  cu.  decimetres  =  i  cu.  metre  (cu.  m.)  =  I  ster  =  i^>  cu.  yd.    " 

ENGLISH   TABLE 

1728  cubic  inches  =  i  cubic  foot  (cu.  ft.)  =  28.3  cu.  dcm.  nearly. 

27  cu.  feet         =  I  cu.  yd.  (cu.  yd.)      =      0.76  cu.  m.      " 
128  cu.  feet          =  i  cord  (cd.)  =      3.6  sters          " 

I.   Volume  of  the  Cube.     Review  what  is  said  about  the 
volume  of  the  cube  on  pp.  15-16. 

1.  What  is  the  volume  of  a  cube  whose  edge  is  5  cm.? 

2.  How  many  cubes  with  an  edge  of  2  cm.  could  be  cut  from  a  cube  whose 

edge  is  10  cm.  long  ? 

3.  Would  it  require  the  same  amount  of  paper  to  cover  the  surfaces  of  the 

original  cube  as  it  would  to  cover  the  smaller  ones  mentioned  in  the  pre- 
ceding question  ?  If  not,  how  much  more  paper  would  be  needed  in  the 
one  case  than  in  the  other? 

4.  How  many  cubes  with  an  edge  of  2  inches  can  be  covered  with  a  sheet  of 

paper  2  feet  square  ? 

5.  If  you  had  a  cubical  block  of  wood  with  edges  u  inches  long,  and  wished 

to  cut  from  it  as  many  cubes  as  possible  with  an  edge  of  3  cm.,  and  to  use 
the  rest  of  the  block  for  cubes  with  an  edge  of  2  cm.,  how  many  of  each 
kind  would  you  get,  allowing  nothing  for  waste  ? 


200 


OBSERVATIONAL   GEOMETRY 


6.  In  the  preceding  case,  if  you  were  to  begin  by  cutting  out  all  the  cubes  you 

could  with  an  edge  of  2  cm.,  how  many  would  you  get  ?  If  you  were  then 
to  use  the  rest  of  the  block  for  cubes  of  the  largest  size  possible,  what 
would  be  the  length  of  their  edges,  and  how  many  of  them  would  you 
have  ? 

7.  Which  will  hold  the  more,  five  cubical  boxes  with  six-inch  edges,  or  six 

cubical  boxes  with  five-inch  edges  ? 

8.  If  you  had  two  cubes  with  edges  four  inches  long,  and  six  cubes  with  edges 

two  inches  long,  how  many  more  of  the  smaller  size  would  you  need  so 
that  you  could  form  a  cube  with  an  edge  of  six  inches  by  placing  them  all 
together  ? 

9.  If  you  have  a  cubical  box  whose  interior  dimensions  are  each  23  inches, 

and  wish  to  fill  it  as  completely  as  possible  with  cubes  of  one  size,  having 
either  a  three-inch  or  a  four-inch  edge,  which  kind  will  leave  the  least 
vacant  space  ? 


2.   Volume  of  the   Parallelepiped.      Review  what   is  said 
about  the  volume  of  the  parallelepiped  on  pp.  29—30. 

1.    How  many  cubic  decimetres  are  there  in  a  chest  which  is  i  m.  I  dcm.  long, 
3  dcm.  wide,  and  4  dcm.  5-  cm.-  deep  ? 


How  much  will  the  chest  hold  ?  " 


VOLUMES  201 

2.  If  the  diagram  on  p.  19  were  folded  up  so  as  to  form  a  parallelepiped,  what 

would  be  its  volume  ? 

3.  How  much  paper  would  be  needed  to  cover  a  parallelepiped  6  cm.  X  3  cm. 

X  2  cm.  ? 

4.  Can  bricks  8  in.  X  4  in.  X  2  in.  be  piled  up  so  as  to  form  a  cube  with  an 

edge  of  2  feet  ?     If  so,  how  many  would  it  take  ? 

5.  How  many  cubes  with  an  edge  of  5  cm.  can  be  cast  from  a  block  of  iron 

25  cm.  X  15  cm.  X  8  cm.  ? 

6.  If  the  cubes  mentioned  in  the  preceding  question  were  to  be  cut  from  a 

block  of  wood  of  the  same  size  as  the  iron,  some  of  the  material  being 
necessarily  unused,  how  many  cubes  could  be  obtained  ? 

7.  How  many  bricks  8  in.    X   4  in.   X  2  in.  would  be  needed  for  a  wall  80  ft. 

long,  6  ft.  high,  and  8  in.  thick  ? 

8.  If  the  wall  mentioned  in  the  preceding  question  belonged   to  a  building  30 

ft.  wide,  how  many  bricks  would  be  needed  for  all  four  walls  ? 

9.  If  you  had  a  block  of  wood  18  cm.  X    12  cm.  X  8  cm.,  and  wished  to  cut  it 

either  into  cubes  with  an  edge  of  3  cm.,  or  into  parallelepipeds  6  cm.  X 
4  cm.  X  2  cm.,  which  would  you  choose  so  as  to  lose  the  least  of  the 
material  ? 

10.  Which  will  hold  the  more,  a  certain  number  of  boxes  each  7  in.  X  5  in.  X 

3  in.,  or  half  as  many  boxes  each  14  in.  X  10  in.  X  6  in.  ? 

11.  How  many  cubic  decimetres  are  there  in  a  chest  which  is  I  m.  2  dcm.  5  cm. 

long,  3  dcm.  5  cm.  wide,  and  44  cm.  deep? 

3.  Volume  of  the  Prism.  Review  the  subject  of  prisms, 
on  p.  34.  The  prism  there  described  has  a  triangular  base, 
equal  to  one-half  of  the  square  face  of  a  cube,  and  a  height 
equal  to  an  edge  of  the  cube.  The  volume  of  this  prism  is 
evidently  equal  to  one-half  that  of  the  cube;  that  is,  the 
volume  is  equal  to  the  area  of  its  triangular  base  multiplied 
by  its  height. 

The  same  is  true  of  any  prism :  the  volume  is  equal  to  the 
area  of  the  base  multiplied  by  the  height.  The  base  is  a 
polygon,  whose  area  can  be  found  by  the  methods  given  on 
pp.  190-194;  if  the  prism  is  a  right  prism,  the  faces  are 
all  rectangles,  and  the  height  of  the  prism  is  equal  to  the 
length  of  the  lateral  edges. 

1.  If  the  diagram  on  p.  33  were  folded  up  so  as  to  form  a  prism,  what  would 

be  the  volume  ? 

2.  The  prism  described  on  p.  123  has  for  its  base  a  pentagon  whose  area  is 

about  10.75  S(l-  cm> 


202 


OBSERVATIONAL   GEOMETRY 


What  is  the  volume  of  this  prism  ? 
What  is  the  total  area  of  its  surface  ? 

3.  Find  the  volume  and  the  area  of  the  entire  surface  of  a  right  hexagonal 

prism,  each  of  whose  edges  is  5  cm.  long,  the  area  of  the  base  being  65 
sq.  cm. 

4.  Find  the  volume  and  the  area  of  the  entire  surface  of  a  right  prism  whose 

height  is  10  cm.,  and  whose  base  is  a  right  isosceles  triangle  with  the  equal 
edges  5  cm.  and  the  longest  edge  7.1  cm. 

5.  Find  the  lateral  area,  the  entire  area,  and  the  volume  of  a  right  prism  whose 

lateral  edges  are  8  inches  long,  and  whose  base  is  an  equilateral  triangle 
with  an  edge  of  2  inches. 

4.  Volume  of  a  Cylinder.  Review  the  experiment  on  the 
volume  of  a  cylinder,  p.  100. 

The  cylinder  there  described  has  for  its  base  a  circle  whose 
diameter  is  equal  to  an  edge  of  the  cube  with  which  the 
cylinder  is  compared ;  and  for  its  height  an  edge  of  the  cube. 
The  volume  of  the  cylinder  is  found  to  be  about  three-fourths 
of  the  volume  of  the  cube. 


The  area  of  the  base  of  the  cylinder  is  about  three-fourths 
of  the  base  of  the  cube:  that  is,  the  circle  is  equal  to  about 
three-fourths  of  a  square  constructed  on  its  own  diameter; 
or,  since  the  square  on  the  diameter  is  four  times  as  great  as 
the  square  on  the  radius,  the  area  of  the  circle  is  about  three 
times  as  great  as  the  square  on  its  radius. 

With  more  accurate  measurements  the  area  of  the  circle 
would  be  found  to  be  nearly  equal  to  -^  of  the  area  of  the 
square  on  its  diameter,  or  2y2  of  the  square  on  its  radius. 


VOLUMES 


203 


There  are,  then,  four  expressions  which  you  may  use  for 
the  area  of  a  circle :  — 

1.  |  of  the  square  on  the  diameter. 

2.  3  times  the  square  on  the  radius. 

3.  \\  of  the  square  on  the  diameter. 

4.  \2  of  the  square  on  the  radius. 

The  first  two  are  sufficient  for  rough  estimates ;  the  last  two  are  accu- 
rate enough  for  any  calculations  which  you  will  need  to  make  in  elementary 
geometry. 

The  volume  of  any  cylinder  is  equal  to  the  area  of  its  base 
multiplied  by  its  height. 


If  the  straight  line  which  joins  the  centres  of  the  bases  of  a 
cylinder  is  perpendicular  to  the  bases,  the  cylinder  is  called 
a  right  cylinder. 

In  this  case  the  lateral  (or  curved)  surface,  as  was  shown 
on  p.  98,  is  formed  by  a  rectangle  having  for  its  edges  the 
circumference  of  the  base  and  the  height  of  the  cylinder. 
The  area  of  the  lateral  surface,  therefore,  is  equal  to  the 
circumference  of  the  base  multiplied  by  the  height  of  the 
cylinder. 

The  length  of  the  circumference  of  a  circle  may  be  con- 
sidered to  be  either  3  times  or  3^  times  the  length  of  the 
diameter,  according  to  the  degree  of  accuracy  required. 


So4  OBSERVATIONAL  GEOMETRY 

1.  Find  in  cubic  cm.  the  volume  of  the  cylinder  described  on  p.  98,  as  hav 

ing  a  diameter  and  height  of  5  cm.,  obtaining  a  more  accurate  value  for  the 
area  of  the  base. 

2.  The  radius  of  the  base  of  a  cylinder  is  14  cm.,  and  its  height  is  10  cm. 

Find,  first  roughly,  and  then  more  accurately :  — 

(a)  The  area  of  the  base. 

(b)  The  area  of  the  lateral  surface. 

(c )  The  area  of  the  entire  surface. 

(d)  The  volume  of  the  cylinder. 

3.  If  you  had  a  block  of  wood  of  the  dimensions  of  the  right  parallelepiped 

described  on  p.  19  (4  in.  X   3  in.  X  z  in.),  what  would  be  the  volumes, 


."  How  much  leather  will  it  take  to  cover  the  vaulting-horse  ? : 


VOLUMES  205 

by  rough  estimate,  of  the  greatest  cylinders  which  you  could  cut  from  it, 
using  for  a  base,  — 

(a)  The  greatest  face  of  the  block. 

(b)  The  second  greatest  face  of  the  block. 

(c)  The  smallest  face  of  the  block. 

4.  What  would  it  cost  to  paint  the  surfaces  of  the  three  cylinders  mentioned 

in  the  previous  question  at  one  cent  per  sq.  inch  ? 

5.  If  you  have  a  cubical  box  whose  interior  dimensions  are  10  inches,  how 

many  cylinders  can  you  pack  into  it,  each  having  a  two-inch  diameter  and 
being  4  inches  tall  ? 

How  much  sawdust  would  you  then  need,  to  fill  up  the  vacant  space  in  the 
box? 

6.  A  party  of  boys  have  a  metre  stick  and  an  English  tape  measure  with  which 

to  find  the  surface  and  volume  of  a  gymnasium  "  vaulting-horse"  which 
has  the  shape  of  a  cylinder  with  hemispherical  ends.  They  find  that  the 
length,  not  including  the  ends,  is  5  decimetres,  and  the  circumference  is  33 
inches. 

(a)  What  is  the  entire  surface  in  sq.  cm.  ? 

(b)  "     "     "         "  "        "  sq.  ft.  ? 

(c)  "    "     "        volume  in  cu.  cm.  ? 
(J)         "     "     "  "       "   cu.  ft.  ? 

5.  Volume  of  a  Pyramid.  —  Review  the  experiment  on  the 
volume  of  a  pyramid,  pp.  63—64. 

The  pyramid  there  described  has  a  square  base,  equal  to 
the  base  of  the  cube  with  which  the  pyramid  is  compared, 
and  a  height  equal  to  the  height  of  the  cube.  The  volume 
of  the  pyramid  appears  by  experiment  to  be  one-third  of  the 
volume  of  the  cube. 

Let  us  now  investigate  the  volume  of  the  pyramid  by  a  dif- 
ferent method.  Suppose  a  pyramid  to  lie  within  a  cube,  the 
base  of  the  pyramid  being  one  of  the  faces  of  the  cube,  and 
the  apex  V  being  at  the  centre  of  the  cube;  the  height  of 
the  pyramid,  therefore,  is  equal  to  one-half  the  height  of  the 
cube.  Now  if  you  imagine  each  face  of  the  cube  to  be  the 
base  of  a  pyramid  having  its  vertex  at  V,  you  will  see  that 
the  six  pyramids  just  fill  the  cube.  The  volume  of  each  pyr- 
amid is  one-sixth  of  the  volume  of  the  cube;  or  one-sixth  of 
the  area  of  its  base  multiplied  by  the  height  of  the  cube;  or 
one-third  of  the  area  of  its  base  multiplied  by  its  own  height. 


206 


OBSERVATIONAL   GEOMETRY 


The  volumes  of  all  pyramids  are  found  by  this  rule:  mul- 
tiply the  area  of  the  base  by  the  height  of  the  pyramid,  and 
divide  the  product  by  3. 

The  area  of  the  base  can  be  found  by  the  methods  you 
have  used  in  finding  the  areas  of  polygons.  The  height  may 
be  measured  by  resting  a  ruler  horizontally  on  the  apex  of 
the  pyramid  and  some  object  which  has  a  vertical  surface, 
and  then  measuring  the  height  along  that  surface. 


If  the  base  is  a  regular  polygon,  and  the  apex  lies  directly 
over  the  centre  of  the  base,  the  figure  is  called  a  regular  pyr- 
amid. The  lateral  surface,  in  this  case,  consists  of  equal  tri- 
angles having  for  bases  the  edges  of  the  polygon,  and  for 
altitudes  the  slant  height  of  the  pyramid. 

1.  What  is  the  volume  of  a  pyramid,  the  area  of  whose  base  is  24  sq.  cm.,  and 

whose  height  is  7  cm.  ? 

2.  In  the  pyramid  described  on  p.  65,  the  edges  are  each  5  cm.  long ;  the  altitudes 

of  the  faces  are  about  4  3  cm. ;  the  height  of  the  pyramid  is  about  4.1  cm. 
What  is  the  volume  of  this  pyramid  ? 
What  is  the  area  of  its  entire  surface  ? 

3.  What  is  the  height  of  one  of  the  six  equal  pyramids  which  will  exactly  fill  a 

cubical  box  whose  interior  dimensions  are  14  inches  ? 

4.  What  is  the  volume  of  the  smallest  cubical  box  within  which  you  could  en- 

close the  pyramid  whose  surface  is  represented  by  the  diagram  on  p.  59  ? 

5.  What  is  the  volume  of  a  pyramid  which  could  be  enclosed  in  the  triangular 

prism  whose  surface  is  represented  by  the  diagram  on  p.  33,  the  base  of  the 


VOLUMES  207 

pyramid  covering  the  base  of  the  prism,  and  the  apex  of  the  pyramid  just 
reaching  the  top  of  the  prism  ? 

6.  If  the  rectangular  parallelepiped  described  on  p.  19,  whose  dimensions  are 

4.  in.  X  3  in.  X  2  in.,  were  divided  into  six  pyramids  of  three  different  sizes, 
each  pyramid  having  one  of  the  faces  of  the  parallelepiped  for  its  base, 
where  would  the  common  point  of  their  apexes  be  situated  ? 
What  would  be  the  volume  of  each  of  the  six  pyramids? 

7.  The  greatest  pyramid  in  Egypt  has  a  base  693  ft.  square  and  a  height  of  500 

ft.     What  is  its  volume? 

6.  Volume  of  a  Cone.  Review  the  experiment  on  the  vol- 
ume of  the  cone,  pp.  106-107. 

The  cone  there  described  has  a  base  and  height  equal  to 
the  base  and  height  of  the  cylinder  with  which  it  is  compared. 
The  volume  of  the  cone  is  found  to  be  one-third  of  the  volume 
of  the  cylinder. 

The  volume  of  any  cone  may  be  found  by  multiplying 
the  area  of  its  base  by  its  height  and  dividing  the  product 

by  3- 

If  a  line  which  joins  the  apex  of  the  cone  with  the  centre 
of  the  base  is  perpendicular  to  the  base,  the  figure  is  called  a 
right  cone.  The  lateral  (or  curved)  surface  is  then  formed 
from  a  sector  whose  area  is  equal  to  one-half  the  slant  height 
of  the  cone  multiplied  by  the  circumference  of  the  base. 

In  the  case  of  a  cone  which  you  construct  from  a  diagram  of  its  surface, 
the  angle  of  the  sector  will  be  given  with  the  diagram ;  but  you  can  cal- 
culate this  angle  directly  from  a  completed  model.  You  will  notice  that 
the  arc  of  the  sector  has  the  same  length  as  the  whole  circumference  of 
the  base,  which  it  joins.  Now  if  an  arc  of  one  circle  has  the  same  length 
as  the  whole  circumference  of  another  circle,  their  radii  must  be  different ; 
the  arc  will  be  the  same  part  of  its  own  circumference  as  the  shorter 
radius  is  of  the  longer  one;  and  the  number  of  degrees  in  the  arc  is 
the  same  as  the  number  of  degrees  in  the  angle  of  the  sector.  Thus 
in  the  diagram  on  p.  103,  if  the  radius  of  the  arc  is  2\  in.,  and  the  radius 
of  the  base  is  i  in.,  i  -f  2^  =  |;  and  ^  of  360°  is  160°,  which  is  the 
angle  of  the  sector. 

If  ycu  test  the  same  diagram  with  the  metric  measurements  there 
given,  you  will  find  that  the  angle  of  the  sector  appears  to  be  161°  instead 
of  1 60°  ;  this  is  because  the  radius  of  the  sector,  by  exact  calculation,  is 
5.6^-  cm.  instead  of  5  6  cm. 


208  OBSERVATIONAL   GEOMETRY 

1.  What  is  the  volume  of  a  cone  whose  height  is  10  cm.,  and  the  diameter  of 

whose  base  is  7  cm.  ? 

2.  What  is  the  volume  of  the  greatest  cone  which  can  be  cut  from  a  cubical 

block  of  wood  whose  edge  is  10  cm.  long  ? 

3.  The  radius  of  the  base  of  a  cone  is  3  in.,  its  height  is  4  in.,  and  its  slant 

height  is  5  in. 
Find    (a)  The  area  of  the  base. 

(b)  The  area  of  the  lateral  surface. 

(c)  The  area  of  the  entire  surface. 

(d)  The  volume. 

(e)  The  angle  of  the  sector  which  formed  the  lateral  surface. 

4.  How  many  cones  can  be  cast  from  a  cylindrical  iron  bar  20  inches  long  and 

4  inches  in  diameter,  the  cones  to  be  5  inches  tall  and  to  have  a  diameter  of 
2  inches  ? 

5.  The  height  of  a  cone  is  12  cm.,  the  slant  height  13  cm.,  and  the  radius  of  the 

base  5  cm. 
Find    (a)  The  area  of  the  base. 

(b)  The  area  of  the  lateral  surface. 

(c)  The  area  of  the  entire  surface. 

(d)  The  volume. 

(e)  The  angle  of  the  sector  which  formed  the  lateral  surface. 

6.  Suppose  a  right  triangle,  with  edges  6,  8,  and  10  inches,  to  revolve,  first  with 

the  shortest  edge,  and  then  with  the  next  shortest  edge,  for  an  axis.  Find 
and  compare  the  volumes  and  the  entire  surfaces  of  the  two  cones  thus 
generated. 

7.  Volume  of  a  Sphere.  The  volume  of  a  sphere  is  nearly 
equal  to  one-half  the  volume  of  a  cube  whose  edge  is  a  diam- 
eter of  the  sphere  (see  p.  114).  A  more  accurate  value  can 
be  found  by  multiplying  the  volume  of  the  cube  by  -|j. 

1.  What  is  the  volume  of  a  sphere  whose  diameter  is  7  cm.  ? 

2.  What  is  the  volume  of  a  sphere  whose  radius  is  5  cm.  ? 

3.  How  many  cubic  miles  does  the  earth  contain,  the  diameter  being  7912  miles  ? 

4.  If  a  cubic  inch  of  iron  weighs  7  ounces,  what  is  the  weight  of  an  iron  ball 

4  inches  in  diameter  ? 

5.  Eight  spherical  glass  globes,  each  having  a  diameter  of  6  cm.,  are  to  be 

packed  in  a  cubical  box  whose  edge  is  12  cm.     How  much  sawdust  will 
be  needed  to  fill  the  vacant  space  ? 

6.  The  diameter  of  the  ball  on  St.  Paul's  Cathedral  is  6  ft.     How  many  cubic 

feet  are  there  in  the  contents  ? 

7.  How  many  lead  bullets   i   cm.  in  diameter  can  be  cast  from  a  cylinder  of 

lead  whose  length  is  14  cm.  and  diameter  35  mm.  ? 
8    Tf  a  spherical  lump  of  putty  whose  diameter  is  8  cm.  were  moulded  into  a 

cone  of  the  same  diameter,  what  would  be  the  height  of  the  cone  ? 
9.    What  is  the  diameter  of  a  sphere  whose  volume  in  cubic  inches  is  the  same 

as  the  area  of  its  surface  in  square  inches  ? 


VOLUMES  209 

10.    If  a  cylindrical  box  has  a  diameter  equal  to  its  depth,  what  part  of  the 
space  will  be  filled  by  the  greatest  sphere  which  will  fit  into  the  box? 

8.  Volumes  of  Irregular  Figures.  The  volumes  of  irregular 
figures  may  be  found  by  experimenting.  For  instance,  you 
may  take  a  jar  whose  volume  can  be  measured,  and  partly 
fill  it  with  water,  noticing  the  level  at  which  the  water  stands. 
Then  if  you  immerse  the  irregular-shaped  object  in  the  water 
and  notice  the  new  level  to  which  it  rises,  you  can  calculate 
the  volume  of  the  object  thus  indirectly  :  the  apparent  in- 
crease in  the  volume  of  the  water  will  be  the  volume  of  the 
object. 

1.  In  a  cylindrical  well  whose  diameter  is  4  feet,  the  water  stands  12  feet  below 

the  brim ;  but  when  a  heap  of  stones  are  thrown  in,  the  level  of  the  water 
rises  to  8  feet  below  the  brim.  What  is  the  volume  of  the  stones  ? 

2.  A  statuette  is  packed  in  sawdust  in  a  cubical  box  whose  interior  dimensions 

are  3  dcm.,  and  the  box  is  exactly  full ;  but  when  the  statuette  is  taken 
out,  the  level  of  the  sawdust  sinks  12  cm.  below  the  top  of  the  box.  What 
is  the  volume  of  the  statuette  ? 


CHAPTER    XXVIII 

RATIO    AND    PROPORTION 

I.  A  ratio  is  the  relation  which  two  things  of  the  same 
kind  have  to  each  other  in  respect  to  size  ;  the  word  means 
"  a  reckoning." 

For  example,  if  the  line  AB  is  3  cm.  long,  and  CD  is  4  cm.  long,  the  ratio  of 
AB  to  CD  is  f . 


Measure  the  following  lines,  and  find  the  ratio  of  the  first 
to  the  second  in  each  case  :  — 


RATIO  AND  PROPORTION  211 


ii  • 

12- 
13- 

14-- 
15  - 
16 
17  - 
18- 
19- 
^20- 


2.    When  two  ratios  are  equal  to  each  other,  they  are  said 
to  form  a  proportion. 

A-  -B 

c D 

E F 

G-  H 

For  example,  if  four  lines,  AB.  CD,  EF,  and  GH  are  3,  4,  6,  and  8  cm. 
long,  so  that  the  ratio  of  AB  to  CD  is  |,  and  the  ratio  of  EF 'to  GH  is 
f,  then  since  f  is  equal  to  f,  the  ratio  of  the  first  two  lines  is  equal  to  the 
ratio  of  the  last  two,  and  the  lengths  of  the  four  lines  form  a  proportion. 

The  proportion  is  written  in  this  form  :  — 

AB  :  CD  =  EF\  GH 

which  means  that  AB  has  the  same  relation  to  CD  as  EF  has  to  GH, 
or,  as  it  is  commonly  expressed,  " AB  is  to  CD  as  EFis  to  GH" 

Again,  suppose  that  two  squares  have  edges  2  and  3  cm.  long;  then 
the  perimeters  are  8  and  12  cm.  long,  and  we  can  say  that  the  perimeters 
are  proportional  to  two  edges  ;  for  8  :  12  =  2  :  3. 

Write  in  numbers  the  proportion  which  exists  between  the 
perimeters  and  two  sides  of  the  following  polygons:  — 

1.  Two  squares  whose  sides  are  I  cm.  and  3  cm. 

2.  Two  squares  whose  sides  are  3  cm.  and  5  cm. 

3.  Two  squares  whose  perimeters  are  8  cm.  and  12  cm. 


212  OBSERVATIONAL    GEOMETRY 

4.  Two  equilateral  triangles  whose  sides  are  5  cm.  and  2  cm. 

5.  Two  rhombuses  whose  sides  are  I  cm.  and  4  cm. 

6.  Two  squares  whose  perimeters  are  16  cm.  and  12  cm. 

7.  Two  equilateral  triangles  whose  perimeters  are  3  cm.  and  12  cm. 

8.  Two  equilateral  pentagons  whose  sides  are  2  cm.  and  3  cm. 

9.  Two  equilateral  hexagons  whose  perimeters  are  16  cm.  and  20  cm. 

10.  Two  regular  decagons  whose  perimeters  are  15  cm.  and  20  cm. 

11.  Two  squares  whose  sides  are  3  in.  and  4  in. 

12.  Two  squares  whose  sides  are  I  in.  and  3  in. 

13.  Two  squares  whose  perimeters  are  8  in.  and  12  in. 

14.  Two  equilateral  triangles  whose  sides  are  4  in.  and  5  in. 

15.  Two  rhombuses  whose  perimeters  are  12  in.  and  16  in. 

16.  Two  equilateral  pentagons  whose  sides  are  I  in.  and  2  in. 

17.  Two  equilateral  hexagons  whose  perimeters  are  12  in.  and  18  in. 

18.  Two  equilateral  triangles  whose  perimeters  are  12  in.  and  18  in. 

19.  Two  rhombuses  whose  sides  are  2  in.  and  3  in. 

20.  Two  squares  whose  sides  are  2  in.  and  3  in. 

Draw  four  lines  whose  lengths  shall  form  the  following 
proportions :  — 

21.  2  :  5  =  6  :  15.  23.   3  :  2  =  6  :  4.  25.   6  :  2  =  3  :  i. 

22.  i  :  2  =  3  :    6.  24.    2  :  3  =  4  :  6. 

Notice  that  in  these  proportions  the  product  of  the  two 
outer  numbers,  called  the  extremes,  is  equal  to  the  product  of 
the  two  inner  numbers,  called  the  means  ;  thus  2  X  15  =  5  X 
6;  1x6  =  2x3,  etc. 

This  is  expressed  by  saying  that  "  in  every  proportion  the 
product  of  the  extremes  is  equal  to  the  product  of  the 
means."  By  means  of  this  rule,  if  any  three  of  the  numbers 
which  form  a  proportion  are  given,  the  fourth  can  be  found. 

Suppose,  for  example,  you  have  the  proportion 

3:9  =  2:.*-, 

where  the  fourth  number  is  missing,  but  is  denoted  by  x.  Then,  by  the 
rule,  3  X  x  =  18,  or  x  =  6 ;  and  the  proportion  can  be  completed  by  re- 
placing x  by  6  thus  :  3  :  9  =  2  :  6. 

Supply  the  missing  number  in  the  following  proportions  :  — 

26.  5  :  3  =  10  :  x.  28.   x  :  8  =    3  :  4.  30.   3  :  7  =    5  :  x. 

27.  6  :  2  =    x  :  3.  29.    5  :  x  =    3  :  6. 


RATIO  AND  PROPORTION 


213 


3.  If  three  lines  are  given,  the  fourth  can  be  found,  so 
as  to  complete  a  proportion  by  the  following  method. 

Suppose  the  three  lines  to  be  a,  b,  and  c  ;  denote  by  ,i'  the 
fourth  line,  which  will  complete  the  proportion  a  :  b  =  c  :  x. 

From  any  point  P  draw  two  lines  PL  and  PM  at  any 
angle  with  each  other. 


Beginning  at  P,  mark  off  on  one  of  the  lines  the  distances 
PY  equal  to  a,  and  PZ  equal  to  b.  On  the  other  line  mark 
off  the  distance  PW  equal  to  c. 

Draw  the  line  YW,  and  draw  ZV  parallel  to  YW. 

Then  the  distance  PV \t\\\  be  equal  to  the  required  line*. 

That  is,  PY  :  PZ  =  P W  :  PV. 

or  a  \b  =  c  \x. 


214  OBSERVATIONAL   GEOMETRY 

Also  if  you'  test  the  lengths  of  YW  and  ZF,  you  will  find 
that  they  are  in  proportion  both  with  PYand  PZ  and  with 
That  is, 

YW:  ZV=PY\PZ 
and   YW:ZV=PW:PV. 


Notice  also  that  the 
angles     PYW     and     3|< 
PZV  are  equal ;   also 
the  angles  P  WY  and 
PVZ. 

You  may  find  this 
question  in  propor- 
tion difficult ;  but  you 
should  master  it,  as 
we  shall  need  it  pres- 
ently in  problems 
about  land  surveying. 

Surveyors  use  this  prin-     33 
ciple  constantly  :  they  find 
the  lengths  of  three  lines 
in  a  proportion,  and  then 
calculate,  without  actually 
measuring,  the  length  of 
a  fourth   line  which   will 
make  the  proportion  com-     34 
plete. 

Find  by  this  method 
the  fourth  lines  which 
will  complete  the  fol- 
lowing proportions:  35< 


RATIO  AND  PROPORTION  215 

.   Mark  off  two-thirds  of  the  line  AB. 
Hint  :  On  any  line,  as  AL,  from  A,  mark  off  the  distance  AX  equal  to  2 

units  of  length  (centimetres,  inches,  etc.),  and  A  Y  equal  to  3  of  the  same 

units. 


37.   Divide  the  line  CD  into  two  parts,  one  of  which  shall  be  two-fifths  of  the 
whole  line. 


38.   Divide  the  line  EFmlo  two  parts,  one  of  which  shall  be  five-eighths  of  the 
whole  line. 


39.  Draw  a  line  7  cm.  long,  and  divide  it  into  two  parts,  one  of  which  shall  be 

three-fifths  of  the  whole  line. 

40.  Draw  a  line  8  cm.  long,  and  divide  it  into  two  parts,  one  of  which  shall  be 

one-third  of  the  whole  line. 

4.    To  divide  a  straight  line  into  any  given  number  of 
equal  parts. 

(a)    With  the  aid  of  a  graduated  ruler. 


-B 


Let  AB  be  the  given  straight  line,  and  5  the  number  of  equal  parts  into 
which  it  is  to  be  divided. 

The  method  of  division  is  similar  to  that  shown  on  p.  26. 

(b)    With  the  aid  of  compasses  and  ungraduated  ruler. 

Let  AB  be  the  straight  line,  and  5  the  number  of  equal  parts  into 
which  it  is  to  be  divided. 

From  A  draw  AX  of  any  convenient  length  and  making  any  con- 
venient angle  with  AB. 


2l6 


OBSER  VA  TIONAL   'GEOMETR  Y 


Beginning  at  A  lay  off  upon  AX  five  equal  spaces  of  any  convenient 
length  ;  let  D  be  the  last  point  of  division. 

Draw  a  straight  line  from  D  to  B,  and  from  tho  other  points  of  divi- 
sion on  AD  draw,  with  the  aid  of  compasses,  lines  parallel  to  DB. 
These  lines  will  divide  AB  into  five  equal  parts. 

(Y)    With  the  aid  of  a  square  or  parallel  rulers. 

Proceed  as  in  (b\  but  draw  the  parallel  lines  with  the  aid  of  the  square 
or  the  parallel  rulers. 

Lay  off  straight  lines  equal  to  the  following,  and  divide 
them  into  equal  parts  as  indicated :  — 


41- 
42- 

44 
45- 
46 
47- 
48 
49- 
50- 


Three  equal  parts. 
Four        " 
Five         "         " 
Three      "         " 
Six  "          " 

Three       " 


Two 
Five 
Eight 


CHAPTER    XXIX 

SIMILAR    FIGURES 

I.  Similar  polygons  have  the  same  shape;  that  is,  one  is  an 
exact  reduced  copy  of  the  other.  Each  angle  and  each  side 
of  one  polygon  corresponds  to  an  angle  and  a  side  of  the 


other.  The  two  angles  which  correspond  are  equal  in  each 
case ;  thus  angle  A  =  angle  F,  angle  B  =  angle  G,  angle  C 
—  angle  H,  etc.  The  equal  angles  lie  in  the  same  order  in 
the  two  polygons;  thus,  if  you  begin  at  A  and  count  around 
the  polygon  towards  the  right,  the  angles  are  equal,  each  to 
each,  to  the  angles  of  the  other  polygon,  beginning  at  .Fand 
also  counted  around  towards  the  right. 

The  sides  which  correspond  are  not  equal,  but  the  lengths 
of  any  pair  are  in  exactly  the  same  ratio  as  the  lengths  of  any 
other  pair;  thus,  if  AB  is  three  times  as  long  as  FG,  then 
EC  is  three  times  as  long  as  GHt  and  CD  is  three  times  as 
long  as  HI,  etc. 


218 


OBSER  VA  TIONA  L    GE OME  TR  Y 


Any  two  pairs  of  corresponding  sides  of  similar  polygons 
form  a  proportion : 

AB  :  FG  =  BC  :  GH 
CD:  Hl  =  DE:  IJ,  etc. 

The  sides  which  correspond  have  the  same  positions  in  the 
two  polygons  with  respect  to  the  equal  angles,  so  that  if  you 
begin  at  A  and  count  towards  the  right,  the  sides  will  corre- 
spond, each  to  each,  to  those  of  the  other  polygon,  beginning 
at  F  and  also  counted  towards  the  right. 

Two  polygons  would  not  be  similar  if  merely  their  corre- 
sponding angles  were  equal ;  for  instance,  a  square  is  not 
similar  to  a  rectangle.  Nor  would  two  polygons  be  similar 
if  merely  their  corresponding  sides  were  proportional:  a 
square  is  not  similar  to  a  rhombus.  Angles  and  sides  must 
both  be  investigated  before  you  can  infer  that  two  polygons 
are  similar. 

Triangles,  however,  are  an  exception.  Two  triangles  whose 
corresponding  angles  are  equal  must  also  have  their  corre- 


SIMILAR  FIGURES  219 

spending  sides  proportional;  and  if  you  find  that  the  sides 
of  two  triangles  are  proportional,  you  can  infer  that  their 
angles  are  equal,  each  to  each.  You  have  already  tested  this 
truth  in  drawing  proportional  lines  (see  p.  213).  In  the  dia- 
gram, here  repeated,  the  similar  triangles  are  PYW  and 
PZV.  The  angles  P,  Y,  and  W  correspond  to  P,  Z,  and  V; 
P  =  P,  Y=Z,  and  W=  V.  The  sides  PY,  PW,  and  YW 
correspond  to  PZ>  PV,  and  ZV  ;  and 


PY:  PZ=  PW\  PV  =  YW-.  ZV. 

2.    To  draw  a  polygon  which  shall  be  similar  to  a  given 
polygon. 

The  problem  of  drawing  a  polygon  which  shall  be  similar 
to  a  given  polygon  is  performed  by  dividing  the  given  poly- 
gon into  triangles  and  then  drawing  a  series  of  triangles 
which  shall  be  similar  to  these. 


Suppose,  for  instance,  that  you  wish  to  draw  a  polygon  which  shall  be 
similar  to  ABCDE,  but  shall  have  sides  only  two-thirds  as  long.  From 
any  vertex,  as  A,  draw  diagonals  to  the  other  vertices,  and  lay  off  on  AB 
the  distance  AX  equal  to  two-thirds  of  AB.  Draw  XV parallel  to  BC; 
YZ  parallel  to  CD;  and  ZW  parallel  to  DE.  Then  the  polygon 
AXYZW  will  be  the  required  polygon  ;  for  its  angles  are  equal,  each  to 
each,  to  those  of  ABCDE ;  and  its  sides  are  each  two-thirds  as  long  as 
the  corresponding  sides  of  ABCDE. 

1.  Name  the  pairs  of  equal  angles  in  the  two  polygons  ;  also  the  pairs  of  cor- 

responding sides. 

2.  How  do  the  entire  perimeters  of  the  two  polygons  compare  in  length  ? 


220 


OB  SEX  VA  TIONA  L   GE  OME  TR  Y 


3.    Draw  a  square  with  sides  5  cm.  long,  and  within  it  another  square  whose 
sides  shall  be  three-fifths  as  long. 


8. 


9. 


4.   Draw  two  rectangles,  one  with  sides  7  cm.  and  3  cm.  long ;  the  other,  simi- 
lar to  the  first,  but  with  sides  five-sevenths  as  long. 


5.  Draw  two  rhombuses,  one  with  sides  4  cm.  long,  and  angles  of  45°  and 

135°;  the  other,  similar  to  the  first,  but  with  sides  three-fourths  as  long. 

6.  Draw  two  parallelograms,  one  with  sides  6  cm.  and  4  cm.  long  and  angles  60° 

and  120°;  the  other,  similar  to  the  first,  but  with  sides  two-thirds  as  long. 

7.  Draw  two  triangles,  one  with  angles  30°,  60°,  and  90°,  and  the  shortest  side 

3  cm.  long  ;  the  other  similar  to  the  first,  but  with  sides  half  as  long. 
Draw  two  triangles,  one  with  a  base  8  cm.  long  and  the  angles  at  the  ends 

of  the'base  40°  and  70°  ;  the  other,  similar  to  the  first,  but  with  the  corre- 

sponding base  three-fourths  as  long. 
Draw  three  parallelograms,  one  within  the  other,  all  similar,  with  angles 

45°  and  135°,  each  having  sides  two-thirds  as  long  as  those  of  the  next 

larger,  the  sides  of  the  greatest  being  9  cm.  and  45  mm. 
10.    Draw  two  similar  pentagons,  each  angle  of  the  first  being  108°  and  each 

side  3  cm.  long;  the  side  of  the  second  being  two-thirds  as  long. 

3.   Areas  of  Similar  Polygons.     If  the  length  of  the  side,  AB, 
of  a  square  be  doubled,  and  another  square  be  drawn  on  AC, 


B      C 


SIMILAR  FIGURES 


221 


the  new  square  will  contain  four  squares,  each  equal  to  the 
first.  If  the  side  AB  be  tripled,  and  a  square  be  drawn  on 
AD,  this  will  contain  nine  squares,  each  equal  to  the  original 
one. 


Likewise,  if  the  sides  of  a  triangle  T  be  doubled,  and  a  new 
triangle  similar  to  T  be  drawn,  it 'will  contain  four  triangles, 
each  equal  to  T;  and  tripling  the  sides  of  T  produces  a 
triangle  whose  area  is  nine  times  as  great  as  the  area  of  T. 


So  also  with  any  polygons  which  are  similar  to  one  an- 
other, as  P  and  L  :  doubling  the  lengths  of  the  sides  makes 
the  area  of  the  polygon  four  times  as  great;  and  so  on. 

This  is  expressed  by  saying  that  "  the  areas  of  similar 
polygons  are  to  each  other  as  the  squares  of  any  two  corre- 
sponding sides." 

The  square  of  a  number  is  the  number  multiplied  by  itself;  thus  the  square 
of  5  is  25  ;  of  7,  49  5  of  8>  64 ;  of  f ,  f  ;  of  f ,  ||,  etc. 

11.  If  you  add  3  cm.  to  the  side  of  a  square  with  an  ed^e  2  cm.  long,  how  much 

do  you  add  to  its  area? 

12.  One  side  of  a  certain  polygon  is  3  cm.  long,  and  its  area  is  80  sq.  cm.    What 

is  the  area  of  a  similar  polygon  whose  corresponding  side  is  12  cm.  long? 


222  OBSERVATIONAL   GEOMETRY 

13.  Two  corresponding  sides  of  two  similar  polygons  are  5  cm.  and  7  cm.  long 

How  do  their  areas  compare  ? 

14.  The  areas  of  two  similar  polygons  are  50  sq.  cm.  and  200  sq.  cm.     A  cer- 

tain side  of  the  greater  polygon  is  six  inches  long  ;  what  is  the  length  of 
the  corresponding  side  of  the  other? 

15.  How  does  the  area  of  the  diagram  of  the  prism  on  p.  33  compare  with  the 

area  of  the  diagram  which  is  required  to  be  drawn  ? 

16.  If  you  were  to  double  the  length  of  every  line  in  the  diagram  of  the  parallel- 

epiped on  p.  19,  by  how  much  would  you  increase  its  area? 

17.  If  you  need  paper  14  cm.  X  10  cm.  to  draw  the  diagram  on  p.  82,  as  di- 

rected there,  what  would  be  the  dimensions  of  the  paper  needed  if  the 
surface  of  the  pyramid  were  to  have  only  one-fourth  as  much  area  ? 

18.  If  the  dodecaedron  on  p.  122  had  an  edge  i  cm.  long,  the  area  of  its  surface 

would  be  about  20.65  scl-  cm-     What  would  be  the  area  if  the  edge  were 
3  cm.  long  ? 

19.  The  area  of  the  State  of  Kentucky  is  about  40,000  sq.  miles.     What  is  the 

area  of  a  map  of  Kentucky  drawn  on  the  scale  of  i  :  200,000? 

20.  The  distance  in  a  straight  line  from  New  York  City  to  the  end  of  Long 

Island  is  about  115  miles.     On  what  scale  is  this  map  drawn? 


CONEY  ISLAND 


Map  of  Long  Island 

4.  Similar  Polyedrons.  Two  polyedrons  are  similar  when 
one  is  an  exact  reduced  copy  of  the  other.  In  such  figures 
the  corresponding  edges  are  proportional,  the  corresponding 
faces  are  similar,  and  the  corresponding  diedral  angles  are 
equal. 

The  entire  surfaces  are  proportional  to  the  squares  of  any 
two  corresponding  edges. 

Let  us  see  how  their  volumes  compare.  If  the  edge  of  a 
cube  be  doubled,  and  another  cube  be  formed,  it  will  contain 
eight  cubes  each  equal  to  the  first.  If  the  edge  of  the  first 


SIMILAR   FIGURES 


223 


cube  be  tripled,  the  new  cube  will  contain  twenty-seven  cubes 
of  the  same  size  as  the  first.  Thus,  multiplying  the  edge  by 
2  makes  the  volume  8  times  as  great;  and  multiplying  the 
edge  by  3  makes  the  volume  27  times  as  great  The  same 
would  be  true  of  any  similar  polyedrons,  whatever  were  their 
shapes. 


This  is  expressed  by  saying  that  "the  volumes  of  similar 
polyedrons  are  to  each  other  as  the  cubes  of  their  correspond- 
ing edges." 

The  cube  of  a  number  is  the  number  multiplied  by  itself  twice  :  thus  the 
cube  of  2  is  2  X  2  X  2  or  3 ;  the  cube  of  7  is  7  X  7  X  7  or  343 ;  the  cube  of  f  is 
I  X  |  X  f  or  Tfi/3,  etc. 


224  OBSERVATIONAL   GEOMETRY 

1.  Wnat  would  be  the  effect  on  the  volume  of  a  cube,  if  its  edge  were  length- 

ened so  as  to  be  five  times  as  great  as  before  ? 

2.  Two  corresponding  edges  of  two  similar  pyramids  are  3  cm.  and  4  cm.  long. 

How  do  their  volumes  compare  ? 

3.  If  the  octaedron  shown  on  p.  120  had  an  edge  I  cm.  long,  its  volume  would 

be  about  471  sq.  mm.  What  is  the  volume  of  the  octaedron  which  the 
directions  say  is  to  be  made,  which  has  an  edge  5  cm.  long  ? 

4.  If  the  icosaedron  on  p.  121  had  an  edge  i  cm.  long,  its  volume  would  be 

about  2.18  cu.  dcm.  What  is  the  volume  of  the  icosaedron  which  is  de- 
scribed as  having  an  edge  2  cm.  5  mm.  long  ? 

5.  If  the  dodecaedron  on  p.  122  had  an  edge  I  cm.  long,  its  volume  would  be 

about  7.66  cu.  cm.  What  is  the  volume  of  the  dodecaedron  to  be  made 
according  to  the  directions,  which  is  to  have  an  edge  about  1.9  cm.  long? 

6.  The   frustum  of   the  pyramid   described  on  p.  82  is  the  part  left  from  a 

complete  pyramid  when  a  plane  has  been  passed  through,  parallel  to  the 
base,  and  dividing  the  lateral  edges  each  into  two  equal  parts.  How  much 
of  the  original  pyramid  was  removed  by  this  plane  ? 

7.  If  the  diagram  on  p.  123  were  folded  up,  so  as  to  form  a  prism,  how  would 

its  volume  compare  with  the  volume  of  the  prism  which  the  accompanying 
directions  describe  ? 

8.  If  Gulliver  was  six  feet  tall  and  his  nose  was  two  and  one-fourth  inches  long, 

and  one  of  the  Liliputians  was  only  six  inches  tall  but  shaped  just  like 
him,  what  was  the  length  of  the  Liliputian's  nose  ? 

9.  If  it  took  128  sq.  inches  of  material  to  make  Gulliver  a  pair  of  gloves,  how 

much  would  it  take  to  make  a  pair  for  the  Liliputian? 
10.    If  Gulliver  weighed  180  pounds,  how  much  did  the  Liliputian  weigh? 


CHAPTER   XXX 

SURVEYING 

i.  Surveying.  Suppose  that  you  were  to  begin  at  one 
corner  of  your  school-yard,  and  measure  the  length  of  each 
side  with  the  aid  of  a  metre-stick  and  the  size  of  each  angle 
with  the  aid  of  a  protractor.  Then  suppose  that  by  stretch- 
ing strings  from  corner  to  corner  you  were  to  divide  the  yard 
into  triangles  whose  areas  you  could  calculate. 


Finally,  suppose  that  you  were  to  draw  on  paper  a  plan  of 
the  yard,  setting  down  your  measurements  and  calculations. 
In  doing  all  this,  you  would  be  making  what  is  called  a 
survey  of  the  yard. 

To  survey  a  piece  of  land  is  to  measure  its  boundaries  and 
angles,  and  to  ascertain  its  shape,  its  area,  and  its  situation 
with  reference  to  other  land.  The  area  is  found  by  calcula- 
tions after  the  other  measurements  have  been  made;  you 
have  already  been  shown  various  methods. 

Although  each  side  and  each  angle  might  be  measured  as 
we  have  imagined  you  as  doing  in  your  school-yard,  such  a 
process  would  be  tedious  if  the  land  were  of  considerable 


226  OBSERVATIONAL   GEOMETRY 

size,  and  perhaps  impossible  if  the  measurements  were  inter- 
rupted by  trees,  houses,  water,  etc.  So  the  art  of  a  surveyor 
consists  in  making  as  few  actual  measurements  as  possible, 
and  in  ascertaining  the  rest  indirectly.  He  does  this  partly 
by  applying  certain  geometric  principles  and  partly  by  using 
certain  instruments. 

The  principles  are  those  of  similar  polygons,  which  have 
already  been  explained  on  pp.  217-221,  and  are  as  follows: 

I.  Similar  polygons  have  their  corresponding  angles  equal 
and  their  corresponding  sides  proportional. 

II.  Triangles  are  similar  in  all  respects, 

(a)    If  their  corresponding  angles  are  equal;  or 

(£)    If  their  corresponding  sides  are  proportional ;  or 

(c)  If  two  corresponding  sides  are  proportional  and  the 
angles  formed  by  those  sides  are  equal. 

A  surveyor's  instruments  are  merely  more  convenient  and 
accurate  substitutes  for  the  metre-stick  and  protractor. 

For  measuring  lines  he  has  a  steel  tape  from  100  to  250 
feet  long. 

For  measuring  angles  he  has  a  transit 

This  instrument  consists  of  a  protractor,  mounted  on  a 
tripod,  and  having  a  small  telescope  for  sighting  distant 
objects.  The  table  of  the  tripod  can  be  adjusted  so  as  to 
rest  exactly  horizontal,  and  two  small  spirit  levels  on  its  top 
test  whether  it  is  in  this  position.  The  telescope  is  pivoted 
over  the  centre  of  the  protractor,  around  which  it  moves 
accompanied  by  a  pointer  indicating  the  angle  observed. 

A  plumb  line  is  attached  to  the  protractor  at  its  centre, 
and  shows  the  point  on  the  ground  corresponding  to  the 
vertex  of  the  angle  observed. 

For  measuring  heights  the  transit  has  a  second  protractor 
which  rests  vertical  to  the  first ;  and  the  telescope  can  be 
made  to  move  on  this  also,  accompanied  by  another 
pointer. 


SURVEYING 


227 


Surveying  Instruments 

Lastly,  the  surveyor  has  a  levelling  staff  to  indicate  on  a 
distant  object  the  point  which  is  on  a  horizontal  level  with 
the  protractor.  This  is  a  rod  about  six  feet  long,  with  a  slid- 
ing disk  which  can  be  adjusted  so  that  its  centre  may  be  in 
the  line  of  sight  of  the  telescope. 

If  your  school  does  not  own  these  surveyor's  instruments,  there  are 
substitutes  which  will  serve  the  purpose  fairly  well. 


228 


OBSER  VA  TIONAL   GEOME  TR  Y 


For  measuring  lengths  you  can  use  a  fifty-foot  tape  measure ;  or  you 
can  make  a  pole,  three  metres  (or  ten  feet)  long,  divided  into  smaller  parts. 
The  pole  can  serve  also  as  a  levelling  staff. 


A  Transit  Board  (two  views) 

A  surveyor's  transit  is  an  expensive  instrument;  but  a  boy  of  some 
Ingenuity,  who  has  an  idea  of  the  use  to  which  his  work  will  be  put,  can 
make  quite  a  serviceable  substitute  out  of  material  which  he  is  likely  to 
have  at  hand.  The  picture  of  such  an  instrument  is  given  here. 

A  protractor  (360°)  drawn  on  paper  is  pasted  on  a  square  board ;  the 
indicator — a  small  stick  —  turns  on  a  screw  pivot  in  the  centre.  The 


SURVEYING 


229 


ights  are  two  nails  and  two  bits  of  zinc  with  narrow  slits,  the  fixed  nail 
>eing  at  o°  on  the  board.  The  plumb  line  and  the  tripod  can  be  added 
without  much  difficulty,  and  then  the  instrument  can  be  used  with  the 
oard  on  edge  for  taking  elevations,  or  fla-t  for  sighting  objects  on  a  level. 
5y  means  of  the  plumb  line  and  the  board  on  edge,  the  top  of  the  tripod 
an  be  adjusted  so  as  to  be  level,  two  positions  determining  the  direction 
f  the  plane. 

2.  Problems  in  Surveying.  We  will  suppose  now  that 
ou  are  provided  with  surveying  implements  —  transit,  tape 
leasure  or  measuring  pole,  levelling  staff,  and  note-book  — 
nd  are  ready  for  practical  work. 

Probably  you  will  wish  to  take  four  companions, — one  to 
old  the  levelling  staff  while  you  use  the  transit,  two  to  meas- 
re  base  lines,  and  one  to  record  the  observations  in  a  note- 
ook.  It  will  be  well  to  have  every  measurement  repeated 
idependently  by  at  least  one  member  of  the  party;  and  all 
lould  work  out  the  problems  which  result.  Make  your  dia- 
rams  and  calculations  as  neat  as  possible. 

We  will  consider  five  problems:  — 

How  to  calculate  the  height  of  an  object  which  stands  on 

a  horizontal  plane. 
,    How  to  calculate  the  height  of  an  object  which  you  cannot 

approach  very  nearly. 
,    How  to  calculate  your  distance  from   an  object  without 

approaching  it. 
,    How  to  calculate  the  distance  between  two  objects  without 

approaching  either. 
.    How  to  survey  a  tract  of  land. 

Every  operation  in  surveying  begins  with  the  measurement 
f  a  base  line.  This  is  the  distance  along  the  ground  from 
he  plumb  line  under  the  transit  to  some  point  where  the 
^veiling  staff  is  placed.  A  surveyor  tries  to  obtain  the 
sngths  of  all  other  lines  by  calculation ;  so  the  measurement 
>f  the  base  line  must  be  made  very  carefully,  as  an  error  here 


230  OBSERVATIONAL   GEOMETRY 

will  be  repeated  and  probably  increased  in  the  rest  of   the 
work. 

A  horizontal  angle  between  two  objects  is  formed  by  two 
imaginary  lines  extending  from  those  objects  to  the  centie  of 
the  horizontal  protractor  on  the  transit. 


Measuring  a  Base  Line 

To  measure  such  an  angle,  the  surveyor  after  ascertaining 
that  his  transit  is  horizontal,  sights  the  objects  in  turn,  noting 
in  each  case  the  number  of  degrees  indicated  on  the  pro- 
tractor by  the  pointer.  The  levelling  staff  is  usually  held  at 
each  of  the  objects,  and  the  disk  is  raised  or  lowered  until  the 
centre  of  the  disk  is  sighted. 

An  angle  of  elevation  is  formed  by  two  imaginary  lines  ex- 
tending from  the  top  and  bottom  of  an  object  to  the  centre 
of  the  vertical  protractor  on  the  transit.  Like  a  horizontal 
angle  it  is  measured  by  sighting  the  top  and  bottom  of  the 
object,  and  noting  the  degrees  indicated  by  the  pointer  on 
the  vertical  protractor. 

In  the  following  problems  the  lower  point  sighted  is  on  a 
horizontal  level  with  the  transit.  The  height  of  the  transit, 
therefore,  is  finally  added  to  the  height  of  that  part  of  the 
object  which  is  obtained  by  calculation. 

1.    How  to  calculate  the  height  of  an  object  which  stands  on 
horizontal  ground. 


SURVEYING 


231 


«iWhat  is  the  height  of  the  tree?" 

The  picture  shows  a  group  of  boys  making  measurements 
from  which  to  calculate  the  height  of  a  tree,  represented  by 
A X  in  the  diagram. 

X 


The  transit  is  placed  in  a  convenient  position  (T  in  the 
diagram)  with  the  plumb  line  hanging  over  B  on  the  ground. 
The  levelling  staff  (CA  in  the  diagram)  is  placed  directly 
under  the  highest  point  of  the  tree,  and  the  disk  is  raised  or 
lowered  until  its  centre  is  on  a  horizontal  line  with  the  transit. 
The  height  CA  is  noted.  Then  the  top  X  of  the  tree  is 
sighted,  and  the  angle  CTX  noted.  The  distance  AB  is 
measured  on  the  ground. 


232  OBSERVATIONAL   GEOMETRY 

These  measurements  are  sufficient  for  calculating  the  re- 
quired height  AX.  They  should  be  recorded  in  a  note-book 
by  the  boy  who  is  acting  as  clerk,  whose  duty  also  is  to  pre- 
pare a  diagram,  like  ACXTB,  for  future  use. 

The  calculation  should  be  made  afterwards  by  each  boy 
independently,  as  follows :  — 

Suppose  the  measurements  taken  to  be, 


CA  (=  TB)  =    4  feet 
AB  (=  CT)  =  25  feet 
angle  CTX  =  39°. 


Draw  on  paper  a  line  ct  representing  CT  on  some  con- 
venient scale,  say  ^ ;  then,  since  CT  is  25  feet,  ct  will  be 
ifa  of  25  feet,  or  3  inches  long. 

With  the  aid  of  a  protractor,  make  an  angle  at  /  equal  to 
CTX,  that  is,  39°,  and  an  angle  at  c  equal  to  TCX,  that  is, 
90°,  and  prolong  the  lines  until  they  meet  at  jr. 

You  have  thus  constructed  a  triangle  ctx  similar  to  CTX, 
and  their  corresponding  sides  are  proportional.  Measure  the 
length  of  ex  and  compare  it  with  the  length  of  ct.  Suppose 
that  ex  is  I  as  long  as  ct ;  then  CX  will  be  |  as  long  as  CT; 
or,  since  CT  is  25  feet,  CX  is  20  feet.  To  this  the  length  of 
CA  (—  4  ft.)  is  to  be  added,  making  the  length  of  AX  24 
feet,  which  is  the  height  of  the  tree. 

In  the  country  or  in  the  city  you  will  occasionally  wish  to 
form  an  idea  of  the  height  of  some  object  —  a  tree,  flagstaff, 
monument,  or  building — without  the  aid  of  any  instrument. 
With  what  you  now  know  about  similar  triangles,  you  can  do 
this  with  some  accuracy,  provided  that  the  object  is  casting 
a  shadow  by  sunlight;  for  close  by  there  will  probably  be 
some  smaller  object — a  post,  for  instance  —  also  casting  a 
shadow.  You  will  estimate  by  the  eye  the  height  of  the  post 
and  the  length  of  its  shadow;  then,  as  the  ratio  of  the  taller 


SURVEYING 


233 


object  to  its  own  shadow  is  the  same,  all  you  will  have  to  do 
is  to  pace  off  this  shadow. 


Measuring  a  Shadow 

Suppose,  for  instance,  that  AB  represents  a  tower,  and  AS 
its  shadow ;  also  that  DE  represents  a  boy  standing  near  the 
tower,  and  DF  his  shadow.  The  triangles  ABS  and  DBF 
are  similar;  so,  if  the  boy  is  5  ft.  tall  and  his  shadow  is  4  ft. 


long,  the  height  of  the  tower  is  five-fourths  of  the  length  of 
its  shadow.  If,  therefore,  a  boy  knows  that  the  length  of  his 
step  is  21  inches,  and  finds  that  he  takes  32  steps  along  the 
shadow,  he  has  the  length  of  the  shadow  56  ft.,  five-fourths 
of  which,  or  70  ft.,  will  be  the  height  of  the  tower. 


234 


OBSERVATIONAL   GEOMETRY 


2.    How  to  calculate  the  height  of  an  object  which  you  can- 
not approach  very  nearly. 

Suppose  AB  to  be  an  object  which  you  cannot  approach 
nearer  than  C. 


• 


Fountain  in  Central  Park,   New  York 

Measure  a  convenient  distance  CD  on  a  horizontal  line  with 
A.  Set  up  the  transit  at  T  with  the  plumb  line  hanging 
over  D.  You 'will  begin  by  sighting  some  point  F  in  AB 
on  a  horizontal  line  with  T,  and  with  the  transit  measure 
the  angle  FTB.  Then  move  the  transit  to  S,  with  the  plumb 
line  hanging  over  C,  and  measure  the  angle  FSB. 

With  these  measurements  you  can  calculate  the  height 
of  A3. 

Draw  on  paper  a  line  st  representing  ST  (  =  CD)  on  any 
convenient  scale,  and  prolong  the  line  towards  x.  With  the 


SURVEYING 


235 


aid  of  a  protractor  make  the  angles  ftb  equal  to  FTB,  and 
fsb  equal  to  FSB.     From  b  draw  bf  perpendicular  to  tx. 

The  triangles  STB  and  stb  are  similar,  and  give  the  pro- 
portion 

ST  :  *SB  =  st  :  sb 


in  which  57"  and  st  are  already  known,  and  sb  can  be  meas- 
ured on  the  diagram,  so  that  SB  can  be  calculated ;  that  is, 

cp  _  sb  X  ST 
^b  ~         st 

The  triangles  FSB  and  fsb  are  similar,  and  give  the  pro- 
portion 

SB  :  FB  =  sb  :  fb 

* 

in  which  SB  and  sb  are  already  known,  and  fb  can  be  meas- 
ured on  the  diagram,  so  that  FB  can  be  calculated ;   that  is, 


FB  = 


SB  x  fb 

sb 


To  the  height  of  FB  thus  found  you  will  add  DT  (=  AF), 
the  height  of  the  transit,  in  order  to  obtain  the  total  height 
of  AB. 


236  OBSERVATIONAL   GEOMETRY 

3.    How  to  calculate  your  distance  from  an  object  without 
approaching  it. 


On  the  Harlem  River 


Suppose  that  you  are  at  the  point  A  and  wish  to  know 
your  distance  from  an  object  X. 


A  B  a  b 

Beginning  at  A  measure  a  line  AB  in  any  convenient  direc- 
tion and  of  any  convenient  length.     Placing  the  transit  at  A 


SURVEYING 


237 
these 


and  then  at  B,  measure  the  angles  A  and  B.     With 
measurements  you   can   calculate  the  distance  AX. 

Draw  on  paper  a  line  ab  representing  AB  on  any  con- 
venient scale.  With  the  aid  of  a  protractor  make  the  angles 
a  equal  to  A,  and  b  equal  to  B,  forming  the  triangle  abx. 
The  triangles  abx  and  ABX  are  similar,  and  give  the  propor- 

tion 

ab  :  AB  =  ax  :  AX, 

in  which  ab  and  AB  are  already  known,  and  ax  can  be  meas- 
ured on  the  diagram;  so  that  AX  can  be  calculated,  that  is, 

AB  X  ax 


4.    How  to  calculate    the  distance  between  two  points  with- 
out approaching  either. 


Suppose  that  X  and  Y  are  two  points  the  distance  between 
which  you  wish  to  find. 

Measure  a  line  AB  in  any  convenient  direction  and  of  any 
convenient  length. 


238  OBSERVATIONAL   GEOMETRY 

Placing  the  transit  at  A,  measure  the  angles  BAX,  YAX, 
and  BA  Y.  Then,  placing  the  transit  at  By  measure  the  angles 
ABX  and  ABY.  With  these  measurements  you  can  cal- 
culate the  distance  XY. 


First  draw  on  paper  a  triangle  abx,  similar  to  ABX,  ab 
representing  A B  on  a  reduced  scale,  angle  bax  —  angle  BAXy 
and  angle  abx  =  angle  ABX.  From  this,  by  measuring  ax 
and  applying  the  rule  of  proportion,  you  can  calculate  the 
length  of  AX. 

Then  draw  a  triangle  aby,  similar  to  ABY,  ab  representing 
AB  on  the  same  reduced  scale  as  before,  angle  bay  =  angle 
BAY,  and  angle  aby  =  ABY.  From  this,  by  measuring  ay 
and  applying  the  rule  of  proportion,  you  can  calculate  the 
length  of  AY. 

Lastly,  draw  a  triangle  yax  similar  to  YAX,  angle  yax  being 
equal  to  YAX,  ax  and  ay  having  the  lengths  previously 
found.  From  this,  by  measuring  the  length  of  xy  and  apply- 
ing the  rule  of  proportion,  you  can  calculate  the  length  of  XY, 


SURVEYING 


239 


5.    How  to  survey  a  piece  of  land. 

Suppose  ABCDE  to  be  the  land  which  you  are  to  survey. 

To  do  this  you  will  find :  — 
(1).   The  lengths  of  the  boundaries. 
(2).   The  direction,  according  to  the  points  of  the  compass, 

in  which  at  least  one  of  the  boundaries  extends. 
(3).   The  sizes  of  the  angles. 
(4).   The.  area. 

Lastly,  you  will  make  a  plan  of  the  land,  and  indicate  in 
one  corner  of  the  paper  the  scale  on  which  it  is  drawn. 


Begin  by  choosing-  a  position  for  your  base  line.  This 
should  be  done  carefully,  so  that  a  single  base  line  may  serve 
for  the  whole  survey.  One  of  the  boundaries  of  the  land  (for 
instance,  AB)  will  be  your  first  choice ;  but  if  the  boundaries 
are  all  very  long,  or  are  otherwise  inconvenient  to  measure, 
you  can  lay  off  the  line  in  some  other  direction,  as  AP. 

We  will  suppose  that  AB  is  the  base  line  and  that  it  has 
been  carefully  measured.  Then,  using  the  transit,  and  taking 
the  ends  of  the  base  line  as  vertices,  measure  the  angles  BAG, 
CAD,  and  DAE  ;  and  ABE,  EBD,  and  DBC. 

Determine  the  direction  of  AB  with  the  aid  of  a  compass. 

With  these  measurements  you  can  complete  the  survey  by 
calculations. 

First,  by  means   of  the  problem  which  tells  how  to   find 


240  OBSERVATIONAL   GEOMETRY 

your  distance  from  an  object  without  approaching  it,  calculate 
the  distances  of  A  from  the  other  corners  of  the  land. 

Next  draw  on  paper  a  diagram  on  some  convenient  scale, 
showing  the  angles  BAG,  CAD,  and  DAE,  and  the  distances 
AB,  AC,  AD,  and  A£. 


Join  the  ends  of  these  lines,  forming  a  polygon  obcde  similar 
to  ABODE. 

Measure  the  sides  of  abcde,  and  by  means  of  a  rule  of  pro- 
portion (see  p.  212)  calculate  the  lengths  of  the  boundaries 
of  the  land. 

Measure  the  angles  of  dbcde :  these  will  also  be  the  angles 
of  the  land. 

Find  the  area  of  abode  by  any  of  the  methods  described  on 
pp.  190-194;  and  by  means  of  a  rule  of  proportion  calculate 
the  area  of  the  land. 


FOURTEEN  DAY  USE 

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